THE UNTVERSITY OT MANITOBA

TOPTCS TN BALANCED T}{COMPLETE BLOCK DESIGNS

by

ROGER A. KINGSLEY

A THES]S

SUBMITTED TO THE FACULTY OF GRÁDUATE STUDIES

]N PARTIAL FULTTLI,ÍENT OT THE REQUIREMENTS FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MATHEMATICS

II]NNIPEG, MANITOBA

October, 19 73 TOPICS IN BALANCED INCOMPLETE BLOCK DESTGNS

By: Roger Alfred Kingsìey .

A dissertation submitted to the Faculty of Graduate Studies of the University of Manitoba in partial fulfillment of the requirements of the degree of

DOCTOR OF PHILOSOPHY e; sls

Permission has l¡ee¡r gralìtcd to tlìe LllÌRARY OF Tlttt UNIVLiR- slTY OF MANlTollA to lend or sell copies of this ttissert:rtion, to' thè NATTONAL LIBRARY OF CANADA to microfilm this dissertation and to lend or sell copies of the filnr, and UNtVURsITY MICROFILMS to publish a¡r abstract of this dissertation.

The author reserves other publication rights, and neithcr tlìe dissertation nor extensive extracts tion it uray be printed or other- wise reproduced without thc author's written penlrissio¡t. To Nancy, Andrew, Paul, and Michael ABSTRACT

In this thesis, geometric methods of const?ucting sJrrunetric balanced incomplete block designs with parameters

(36, 1s,6), (45, !2, 3'), and (56, !!,2) are described.. other methods of constructing designs with the same Þarameters ane surveyed. Methods of testíng designs for isomo::phism ane gíven, and these methods applied to the designs which we have l-isted. An appendix is included, which contains computen programs for construction of incidence matrices for the designs, programs for conducting isornor phism tests, listings of the block designs, and the results of the isomorphism tests.

(i) ACKNOI^ILEDGEMENTS

I woul-d like to thank Professor R. G Stanton for hls advice and gui.dance during ny graduate program. I l¡oul-d

also like to thank Professors I'1. D. ltallis and H. C. Willians and Mr. Barry Wolk for theÍr advÍce and encouragnent. Finall-y, I would like to thank Miss Sandra Gratton and Mrs. Mary Berthelette for their excellent typing of the several drafts of this nanuscrlpt.

(ii) TABLE OT' CONTENTS

PAGE

ASSTEACT i

ACKNOWLEDGE}ÍENTS ... l-1

CHAPTER ONE - InLroducÈion . I

51.1 - Balanced Incomplete BJ-ock Ðesigns . I 51.2 - ?rojecÈive Geometry 1l

CHAPTER Ti{O - A Design from a FÍnite P1aûe ...... 16

52.1 - Arcs and Ovals ...... 16 92,2 - The Oval-s of Pc(2,4) ...... 25 52.3 - The Block Desígn (56, 11, 2) ...... 33

CHAPTER TIIREE - T¡.zo Cubic Surface Designs ...... 36

53.1 - Prelirninary Topics ...... 36

53.2 - The Double-Six Configuration ...... 39 .53.3 - The StrucÈure of the Conplete Double-Six ...... 43

53.4 - The TrÍtangent Planes ...... 5L

(i1i) CHAPTER I'OUR - Isonorphisn of Designs .... 55

S4.l - The Scith Normal Form ...... 55 54.2 - Ge¡eraLized Intersection Nr¡mbers .. 67

CIIAPTER IIVE - AJ-gorithnic Techniques in BLock Designs . 74 55.1 - ConpuÈer Construction of Incidence Matrices ...... 74 55.2 - Other Constructíons for the Designs ... 80

55.3 - Isomorphisrns among the Designs ..,. 87 55.4 - Isonorphism of RK45, R345, and SS45 ...... 91

APPENDICES

EIBT,IOGRA?IIY

(iv) Chapter I - InÈroduction

The object of Lhis thesis is tr¡ofo1d. First, we shall" present some geometric constructions for balanced incom- pl-eÈe block desÍgns. Second, Ìre shall describe some methodg of tescing different designs for Ísonorphism, and apply these

[eÈhods to several sets of known designs. In this chapter,

!,7e define balanced incomplete block designs and give sone elementary results here as well.

The tåteríal in this ehapter is elementary, and nay be found in a vaÍieÈy of sources, In particular, Ifal-1 [18] and Ryser [30] discuss boÈh BIBDTs and finite geometries. Stanton 1373 provides an excel"lent source for some of the relevant aspects of CombinatorÍal Theory.

$l.l - Balanced IncompleÈe 8lock Designs

A balanced incomplete block design (BIBD) I.r:ith palameLers (v, b, r, k, À) is a pair of sets,

= {Vr Vr, ..., Vrr} ,

and B = {Br, BZ, ..., , "t} calJ-ed, respectively, varieÈies and blocks, and an incidence relaÈÍon between varieties and blocks such that (i) each variety is incidenÈ with r blocks, (ii) each block is incidenÈ a'ith k varíet1es, aûd (fii) every pair of distÍnct varietíes ls nutually incident v¡ith À blocks.

Fron Èhese definiÈions, it is imediately apparent that v>k>0 and b2r>l>0. It is usual, in order that trivíal cases may be avoided, to deroand that these ÍnêqualÍties be sharpened to v> kt 1, and b> r> À> 0. lle are led irûnediately to ask whether a block design exists for any parameters (v, b, r, k, À) satisfying the specÍfied inequal-itÍes. That sone further restri.ctions are necessary is seen fron the following lrell-kno!,m theorem.

Theoren 1.1.1. If a balanced incompleÈe block design exi.sts t/ith parameters (v, b, r, k, l), then (1) vr = bk , and (ii) r(v-r) = r(k-l) Proof: (i) tle shall courÌt the nunber, x, of pairs (V.,, B=) of r-' J a variety and a block r¿hÍch are incídent. First, for a fíxed variety V, , there are r bl-ocks BJ incident with V. . As there are v varieÈíes, we have x = vr. Similarly, by countÍng varieties incident Ì,r1th a fixed block, we obtaÍû x = bk. Hence vr = bk. (ii) In this case, It'e count Èhe nunber, x, of pairs

(V.r,rJ B.) of a variety and a block r"¡hich are incídent, and such 3 that B. is aLso Íncident r.¡ith a fixed varÍety, Vl, ¡lrst, 1et V, be any variety other than Vl. There are I blocks B. incÍdent vith both V. and Vr. As there are v-l choices for Vr, we have x =. À(v-l). 0n the other hand, l-et B. be a fÍxed block incÍdeflt

!,7ith Vl . There are k varieties incident r,rith B . , one of !¡hl.ch Ís V1. Thus, fron 8., ve rr,ay select V. 1n k-l ways.

As there are :c cholces for 8., r^re have x = r(k-l). Hence = r(k-r). I ^(v-r) There are other restríctions on the parameters.

Before r,¡e obtaln these, !,re Íntroduce the idea of an lncídence natríx for any relat.ion on finite sets. Let us take the finlte sets U = {vf, ..., Vr}

and B = {Bf, ..., u¡} , and 1et a ïelation fron U to B be given. Let A = (a..) be the v x b natrÍx given by

(t if V.i and B,j are incident, r'J "-.=l I o oÈhen¡ise. I{e say thêt A is Èhe incídence måtrix for the relatlon. (If U = ß, we shal1 al-ways take V. = Br, t = t, ..., v.) Obviously, the u¿tríx A is dependenÈ upon the order in ¡¡hÍch

¡¡e number Èhe varieËies and the blocks, different numberings leading Èo different mêtrices. We shall return to thÍs pro- blen later, when nre di.scuss isomorphism of designs.

Throughout this thesÍs, we shaLl- Let In denoÈe the n x n idenÈity mátrix, and Jro denote the n x n natrÍx (aij) given by o¿j=l' i = 1, ..., rn J = l, ..., n

Whenever Lhe context nakes their values c1ear, we sha.11 onit the subscripts.

Theorem 1. 1.2 -

(Í) AAT = ÀJ + (r-t)r (1i) Jt.rA = Hlb

Proof: (i) The el-êment b,, in B AAT is the inner product of aJ = ror¡ i and row j of A. These ror¿s correspond to varieÈies V. and V, in the design, and a typical- tern in the inner producÈ -t t¡i1l be I if the varieties are both lncldent !ùith the appropri- ate block, 0 otherwise. Thus b,, counts the number of bl-ocks r--'l rûÍth rrhlch V.r-J and V, are incident. For í = J, thÍs value is r, and for i * j, the value is ),. Thus, B contains r on its diagonal, À el"sewhere.

(ií) JlvA conputes the col-umn sums of A. Each colurnn represents a bJ-ock, and conÈains k lrs. E

Theoren 1.1.3, If a v x b matrix A exists which contains onl"y Ors and lrs, and rnrhich satísfies (i) and (ii) of the previous theorem, then there is ê BIBD r{¡ith parameters (v, b, r, k, À).

We can use Theorem I.I.2 to obtain another testriction on the parameÈers, kno¡vn as ÏÍsherts inequal-ity.

Theoren 1.1.4. bàv ¿nd r>k.

Proof: Since vr = bk, Èhe trrro Ínequalities are equivalenÊ. 5 Now, AAT contains r on its diagonal, L elsewhere, and ís v x v.

IÈ is easy to compute aet (elr) = (r-),) v-l(r (v- l) + r) v-l1r = 1¡-r,¡ (k-r) + r) = rk(r-r)v-l As we have denanded that ê11 parameters be positive and that r > À, vre see that det (AAT) > 0 Thus, AAT is nonsingulâr, and consequently, v = rank (n¡,T) < rank (A) < rnin (v, b). Thus v lb. I thus we see that, if b > v, AT can not be the natïlx of a BIBD. l[e shal1 now prove that, in the case b = v, the exact opposite is true: AT do"" yield ã BIBD.

If a balaneed lnconplete block design has v = b, r,/e shaLl call Èhe design synmetric.

Theoren 1.1.5. For a s)rnnetric BIBD, (1) r=k, v_L 2 (ií) ldet al = k(k:À) , and

(iii) lre = ¡¿r .

Proof: (i) fIe have vr = bk and v = b. Ilence r = k. (ÍÍ) Since A is square, det A exists, and det A = det AT.

Thus, (det A)2 = A"t A det AT = ¿et (aAT) = rk(r-t)v-l = r2(t-r)o-r v-l . Eence laet el = t(k-r)T.

(iii) Let J = Jw. Let k-), = n. lIe have AAT = nI + ÀJ = B¡ and JA = kJ . It is clear 6

that r,7e also have AJ = kJ By (ii), A Ís non- singular. Frorn AJ = kJ , rr,e derive J = A-1 '(kl) or A-J=k*¡-1 -l No!¡ ^T - ^-r^^T =A'(nI+IJ)-l = n¡,-1 '+ LA-l 'J = nA'+-t ìk -l'J Thus lT¿, = oE-le + lk-lJA lkJ = nt + ),k- =nI*trJ=B E

$Ie usual.ly write (v, k, l) rather than (v, v, k, k,

),) f or the parameters of a symetric BIBD.

Corollary 1.1.6. If A is Èhe Íncidence matrix of a s)'metric

BIBD with parameËers (v, k, l), then AT Ís also Èhe íncidence natrix of a syBnetric 8IBÐ r,rith parameters (v, k, I).

The design given by AT is called Èhe duât d.esign of that given by A.

Suppose that a design ¡,/iÈh paraDeters (v, b, r, k, l,) has an incidence natrlx A. We noted earlier that the matrix A depends upon the order in which varieties and blocks are nunbered. Clear1y, renunbering varieti.es is equÍva1enÈ to permuting the rows of A, and renumberÍng blocks is equivalent to pernuting colunns. Clearly, any one of the fanily of

ûatrices obt"io"d fron A by ror,¡ and columr permutations vi1l serve ês an incidence maÈrix for the design. 7 Let D and Dr be two designs with the saDe parameters

(v, b, r, k, I). Let the varieties and blocks of D be U and B, and let the varieties and blocks of Di be [/' and 8r. I'Ie wi]-l say that D and Dr are ¿eggglnþig Íf and onLy if there exists a pafr of bij ections

f : V -+ Vl and g: 8+B' such that V is incÍdent with B (in D) if and only if f(V) is íncidenr wirh g(B) (in Ð').

Lemna 1.J.7. ],et D and Dr be two BIBDTs with the same para- meters, and 1et A and Ar be incidence rnatricês for the designs. Then D ís isomorphic to Dr if and only Íf A' can be obtained fron A by row and coJ-urnn pernutations.

Lenrma 1.1.8. If a symeÈric design is given, then its dual design is uníque up to isomorphism.

There are very strong condi.tions ¡¡hÍch nust be satisfied by the pararneters (v, k, À) for a s)rmetric design.

Theorem 1.1.9. If a s)rrnnetric BIBD has parameters (v, k, À), then

(i) À(v-1) = k(k-l) (ii) If v=0(2), then n = k-À is a perfect square.

Proof: (i). I,¡e have ),(v-l) = r(k-l) and r = k. Our result is imediate.

(ii) Let A be Èhe incidence må.trix for our design. I We knor¿ thaE det À ls an intêger, and have computed /2 laet 'll = o o(v-l) Thus, when v ís even, v-l ls odd, and n must be a perfect square. E

The second part of Theorem l,l.9 is half of the well-knor,m Chowla-Ryser Theorem. The other half of Èhis theorem (c.f. Ryser [28]) deals Írirh the case v = l(2) and asserLs

Theoren l.l.l0. If a s)rmetric BIBD has parameters (v, k, À), v 1(2), then = the equation (v-l) t2 = o*2 + {-g ¡z ^ "' has a non-trivial solutlon in íntegers. (Again, n = k-À.)

Íle note that, at the present time, no counter- example has been obÈaíned for the hypothesis that the condj.tions of Theore¡ns 1.1.9 and 1.1.10 are sufficieût to guarantee the exÍsÈence of a (v, k, À) design.

Lemtra l.l.l1. The paramerer sets (36, 15, 6), (45, 12, 3), and (56, 11, 2) all satisfy the conditÍons of Theorem 1.1.9 or l. l. 10.

Proof: In each case, À(v-l) = k(k-f) ho1ds, and n = k-À = 9. tr'or (36, 15, 6) and (56, 11, 2), we are done when rre note rhat n is a perfect square. For (45, L2, 3), Theoren 1.1.10 rnus t be applied, and ¡'¡e see that (x, y, z) = (1, 0, 3) is a non- trivíal soLution to the Ðiophantine equation. E

We shaLl construct designs with these parameters. 9

There is one meLho4 of presenting BIBDTs which a1l-or¡s a concise description of Èhe design, and a short nethod of test_ ing that the purported design exists. This is Èhe construc- tion by means of a group d.ifference set.

A group dÍfference set (or difference set) with Parameters (v, k, À) is a k-subset le, e2, ..., B¿Ì of an abelian group.G of grd.er,v_such that if d is a non-zero element ôf G, there are exactJ-y À ordered. paírs (g,, -1- -J-g,) such that d = Cr - gj'

A sÍuple,examptre is afforded by Il,.2,4] in the group of lntegers nodulo 7.. ihis.is a (7,3; l) difference

set, !,le shal-l- aLlow one abuse of staûdard maÈhema.tical nota_

tion, which is best illustrated by an exanple. It is r^rell_

knoarß thaL a (16, 6, 2) dÍfference set is given by {(r,0,0,0), (0, 1,0,0), (0,0, 1,0), (0,0,0, r), (1, 1;0, 0), (0, 0, I, l)] ín rhe group C2& c.-s C2ø C2.

tr{e shall- abbrevi-ate Èhts by saying Èhat

{lOO0, 0100, 0010, 0001, 1100, 00tt nod (2222)} is a

(16, 6, 2) dífferenee set. This noÈaÈion ÈacíÈly assumes

that the abellan group G fs the dLrecË sr¡m of cyclic groups of orders < 10.

Theoren 1.1.12. If a (v, k,. À) difference seÈ exísÊs, Ëhen there is a s)¡metric BIBD wÍth parameÈers (v, kr. À).

Proof: LeË G be an abelian group of order v, and Let K be the k-subset of c which yiel-ds a (v, k, l) difference set. Let ür = tvglg e cÌ and I = {Bglg e G} be two t0 v-sets of fonnal synbols. We define an incidence reLaËion

Vg ís incídenr with Bgr if and only if g - gr € K. We norr prove that $re have defined a (v, k, ),) 3IBD.

Let K = {gl, ..., gL}. Fix a varÍeËy Vg. Now, the¡e are exactly k choices for gr such Èhat g - g, e K, namely, g' = g - gi , I < i < k

Thus, Vg is incident with exactly k blocks. Sinilarly, Bgr is lncident v/ith k varieties. Tínally, 1et Vg and Vgr be tI'ro distincÈ varieties. These varieties occur together ín Bgrr íf and only 1f 8-g"=aeK

and gt - gtt = b e K . Subtracting, ve have 8-91 =a-b'

Now, the dj-fference d = g - Bt is fixed, and non-zero. Hence, there are exactly I distinct values for g',.

I,Ie make one coment: if r"re def ine incidence by rrvg I is ÍncidenÈ rtr'iÈh Bg if and only if g + g' € K ", the theoren remains vaIid. The two designs thus obtained are isomorphLc (cf. Theorem 4.2.4). 11 51.2 - Projective Geometry

In this secti-on, we define a projective geometry,

and present a faniLy of explicit exampJ"es of projective geo- netries. Irlhile this fani.J-y r,riLL not e:(haust all projective geometrLes, it is sufficienÈly large for our purposes.

A pro.iective geonetry i-s a syster composed of tr,üo disjoint sets, P and L, ca11ed poj-nts and Lines respéctiveJ-y, and a relation I c P x L cal-led incidence, which saÈisfies

certain axioms; .Before preêenÈlng Lhe axior0s, we introd.uce

some optlon:al tereinology foï the incidence relation: if the

point P and the Line g aíe incident, we nay also say t'P is on 4tt, tt.& passes through p", or even "0 contains pr.

If there is a poinÈ P incident rûith two distlnct lines k and I'k [, we will say that neets .Ctr or, ûore exp]-iciÈ1y, that "k meets l, in Ptr. The axloms r¡trich our objects r0ust satísfy are Èhree in number: (41)" There are at l-east four dístÍnct points, ¿nd no thrêe of these are i.ncident with a single J_ine.

(42) Every pair of distincÈ points is contained in exactly one Line. (The points A and B Lie on a lÍne denoLed by AB.) (43) If A, B, C, D are four dÍstirict points, and if AB meets CD, then AC meeÈs BD. (À4) Every l-íne is incident ¡rith at leasr Ëhree distÍnct poin¿s.

Our definitlon nakes it cl-ear thaÈ a line and Êhe set of poÍnts on thaÈ 1íne are dísttnct entÍtíes. I{e shall r generally not disÈingulsh betl,reen Èhem, allowlng a'r cêrÈaf¡l T2 a¡nount of set terminology aq a coûvenience.

" ProjectÍve geonetries may be derived from vector sÞaces; so are present a mÍninal amount of linear algebra here, withouË proof.

Let T be a fieJ-d, and Let V be a vector space

over tr'. Underscored Roman letters denote vectors.

i) If X c U , S(X) is thê smalLesÈ subspace of conÈaíning X.

11) rf Ul arrð, U, are subspaces of y, then Ut + Uz = s(úlr u.U2).

11i) dÍû ¿It + dln ü, = U'r (Ut + IJZ) + din (U, n f.lr) iv) s(x) + s(Y) = s(x.u y)

Theoren 1.2.1. Let V be a vector space of dimension 3 or greater over F. If ¡ve caLl l-dimensÍonal subspaces, points, and 2-dinensÍona1 subspaces, Línes; and if rse define ,

incidence by containmenË, then we have a projective geomecry.

Proof: Si-nce U has d.ímension at l-east 3, there are at l-east 3 j-ndependent vecÈots in U Let el, g2, e, be Índependenr. LeÈ f = 91 + e2 + 93. It is cl-êar rhar S(Cr), S9), tG3) and S(f) are 4 di.srlncr poinrs, and rhar any 3 of the points generate S(.9t, 92, g.3) .; which has dinensi-on 3. Thus, no 3 of the points lie on a single 1ine, and (41) holds. Secondly, rhe poinrs S (g) and S(Ð are dístfnct i.f ancl onJ-y Íf {u, v} is an índependent set. In rhis case, S(u, v) Ís the rmÍque 1i-ne contatnLng S (g) and S(v) 1-3

Thls veriftes (42). Third,.let Sqà), í = 1, ..., 4, be disrincr points. rf s(ul, !2) rnee ts sG:, gq), theÍr inrersecËion has dimension > 1, and dirn S(grr !2, 13, g4) = 3. If, on the oÊher hand, S(grr 93) ânat S(g, g,_) do nor meer, din S(q, .1f2, 13, $) is 4o a conËïadicÈior. Thus, (A3) holds. trinally, to see (A4), we noÈe thaÈ the 1íne S(u, v) is incÍdenÈ r,zith the poinÈs SG), S(v), and S(g + \¡). Thus, we have a proj ective geonetry. g

If U Ís the vector tr'o+l (n > 2), rre d.enote the "p"". geometry of Theorem 1.2.I by pc(n, ¡') . If, furrher, F = GÌ'(s) r¡here s is a primê poÌrer, !¡e $/íLL Ì'rlte Pc(n, s).

In the vector It+l every "p"". , vector is an (n*1)- tup1e. A point in the corresponding geomeËry is a 1- dirnensional subspace, and this Ís the set of all scalar nultiples of any non-zero vector in Ëhe subspace. hle agree Èhat any non-zêro vector shall serve as coördinates for its corresponding poLnt, From a set of co'órdÍnates for a particular point ray be obtained other coö¡dÍnates for the same poÍnÈ, by nul_tiplyíûg by any non-zero scalar. Io particuLar, in Pc(n, s) , every poinr has (s-1) dÍstinet coöidinate sets. are called hgnggeneous coördinates. .These In PG(n, F), an r-flat is the set of poinËs in an (rtl)- dimensional subspace of the underlying space I'n+1 . This has meaning for -l < r < n , al-though a (-l)-fLat is the nul"L seË. In this terninology, the poi¡,ts are o-flats, the lines are l-fl-aÈs, and there is exacËly one n-flat, namely the set L4 of all point.s Ln the geometry. Ior PG(n, s) , we can count various quanÈitíes associated rrÍth r-flats. rl-l Theoren 1.2.2. In ?G(n, s) any r-f1at contai"" , * s- r poínts .

Proof: lle are, in fact, counting the numbe¡ of points in an

(rl-l) -dinensional- subspace. There are sÉl vectors in Èhe subspaee, of which one is the zero vector. The remaining vecÈors deternine each point s-l tlmes, hrhence the number rl-l . s -l or DoLnEs- Ls ------s-r . E

Corollar-y 1.2.3. The nuuber of points ín PG(n, s) ls

("o*1-r) / ("- r)

Coroll-ary 1.2.4. The number of points on any line (1-f1at)

Ls s+1. As we shall be particularly interested in planes (2-flats), hre note

CoroLlary 1.2,5. Pc(2, s) contains I + s + s2 points. There are I'dua1'r results for coroll-aries I.2.4 and 1.2.5 in the sense of llall [17],

Theorem 1.2.6. In PG(2, s), any two lÍnes neet.

?roof: Let S(u,, u^) and S(u^, .-{tu,) be lines. SÍnce Èhe -t -z -J underlying space has dimension 3, then .vector dtn S(gr, llZ, 9¡, +) . 3 , whence dün (sG' n2) n S(u3, go)) I l, . and there Ls êt l-easÈ a poinÈ in comon. E 15 CoroLlary L.2.7. Through every point pass s+1 lines.

Proof: Lêt P be a point, and let l, be a line not through P . Every line through p neets .1, in a point, and every poinË of .1, determ.ines a 1ine through p . Thus, the nunber of lines through p is equal to the nuiBber of points on .{. , nameJ.y s*l . g

CoroLlary 1.2.8. There are I * s * s2 lines in ?G(2, s).

Lenma 1.2.9. PG(2, s) is a s)¡nnetric BIBD wÍth parameËers (s'+s+a I, s+1. l). t6

Chapter 2 - A Ðesígn from a I'inite Plane

52.1 - Arcs and Ovals

ln this section, rüe discuss Èhe existence of sets of ,. . poÍnts ln a finite plane, no three collinear. Upper bounds on

the slze of such sets are obtained, and the number of such sets counted for certain values. Thê results 2,1.1 - 2.1.8 are originally due to Bose [7]. The geonetrlc proofs coüe fron Quist [29J. The counting theorems 2.f .9 - 2.1.!4 axe due to the author. Consltler the finÍte projective plane ?c(2, s), r,'here s ls a prime por^rer. I'Ie say that a set of points is an arc if

no three of thê poínts are collinear. More speclfi-cally, an arc contaÍning D poÍnts is an m-arc.

Tr¡o numbers associated r.¡ith the arcs of PG(2, s)

are of Ínterest. First, r{e shouLd like Lo knon thê greatest ioteger n = n(s) for which an m-arc exists. Second, we :.:..:t:l shoul-d like to knor¡ how many k-arcs there are, for I < k < n(s). [Íe shall call this nunber n(k, s) . I'inal1y, an n(s)-arc is

termed an ova1. L7 the nunber n(s) has been studied in a more general context by Qulst [29] and others, Let s be a prime power, and f,/ the vector space of dimension r over GF(s) Then n(t, r, s) Ls used to denote the cardÍnality of the largest set of vectors in U rrith the property thaÈ any t are Índepeodent. It is evident that our quantÍty n(s) is, in fact, n(3, 3, s)

Theorem 2.1.1. n(s) < s+2

?roof: Let A be an m-arc Ín ?G(2, s) , and let P be any point of A . There are s+l liDes through A , each of r¿hich may contaírr at most one further poínt of A . Thus mll*(s+1)=sa2

Theoreu 2.1.2. s+l < n(s)

Proof: LeE A = {(0, o, 1), (0, r, 0)} u {(1, e, c-1)lc e GF(s) and a # 0}

A is clearly an (s*l)-set, and we need only tesÈ r,rhether A is an arc. I{e may tes! any three points of A by the deter- minant test. There are four cases. Case (i) ConsÍder the poinËs (1, c, a-l), (0, 1,0), (0,0, l).

I'le have lr o o-tl lo I o l=rlo, lo o I I so the points are ûot collinear. t8 Case (ii) considêr the polnts ¡t, c, a-l), .(f, Ê, Ê-1), (0, 1, 0), where a # ß We have o o-1 ll | It B ß-rl=o-t-s-l #0. lo r ol Again, the points are not collinear.

Case (iii) conslder rhe poinrs (1, o, o-1), (1, Ê, B-1), (0, 0, f), rnrhere c t ß . I,Ie have Id'd -t I ß g'-t =ß-c#o 001

Again, the points êre noÈ collÍnear.

Case (1v) Consider the polnts 11, c, a-l), (1, ß, ß-1), (f, f, y-1), Ìrhere c, ß, y are all distinct. I,Ie have ll a cr-1 ^ -1- It u ß-r = (a-ß) (e-r) (r-d) (aßr) , lt , Y-' r"¡hÍch Ís non-zero, as each of its factors is non-zero. Again, the poÍnts are not collinear.

As this includes aIJ- possible cases of three points fron A , we conculde that A is an (s+l)-arc, and. that s*l < n(s) a

¡efore *e proceed to deternine r¿hich of s+1 or s*2 is the correct value for n(s) , we shal1 define some additional tårms. Let A be an arc, and. .C any 1ine. As

D'o three points of A are coLl-inear, the line .C úeets A in 0, l, or 2 poinÈs. trüe shaLl say that .C is a skew line, L9 a or a decarit .taûgenL, to A , accôrding as the number j.s 0, 1., or 2.

'Tþeoren 2.1.3. An (s*l)-arc has a unique tangent at every point. Proof: Let P þe a Foint of the (s+l)-arc A . There are s+l

J-ines through P , s of whlch are secanËs (one for each

renaining point of A). This 1eaves one and onJ-y one f-ine through P , which forms a tangent. E ÍIe note rhat, if an (s*2) -arc exÍsÈs, each of its (stl) -subsets ís an (s*l)-arc.

. An (s+l) -arc nay be exrend.ed to an (s*2)-arc Ín aË nost one nay. This extension Ís possible if and only if the tangenÈs of the (s+1) -arc are concurrent.

Proof: Suppose the tangents to an (sfl)-arc, A, concur in the point P . Then Ar = A u {p} is an (s*2)-ser. Now, there are s+l tangents to A , a1I passing through p . Thus, no secants of A pass Èhrough p , and no Èhree points of Ar nay be co11inéar, so A, is an (si2)-arc.

Second, suppose A' = {p} u A is an (s.t2)-arc. The lines fron P to points of A are clearly Èhe taogents to A; so these tangenËs concur in p . As there Ís aÊ most one poÍnt

? in lrhich al]- tangenÈs may concur, there is at mosË one . extension Èo (s*2) of A an -arc. g .

Theoren 2.1.5. If s is even, the Èangents to any (s+i)-arc are concurrent, and the arc can be extended Ëo an (s+2)-atc. 20 Proof: Let A be an (s*l)-arc, and 1et ? and Q be distínct points of A . The line ?Q is a secant. Let X be any point

on the lÍne Pq . There are a mruber, say n, of secanÈs through X. There are then 2n points of Ä lying on secants through

X. llowever, s*l is odd, so there is at least one point, Y , ¡shich 1s in A , ûot on a secant through X . This neans that

the line XY Ís a tangent. Thus, through every point on a

secant passes aË least one tangent. There are s*l points on

any secant, and s+l tangents. Thus, through every point oo

a secant passes exactly one tangent. If P is a poinÈ where any trúo tangents meet, there are no secants through P , whence the (s+1) lines joining P to poinrs of A are all the Ëan-

gents to A , and A u tP] 1s an (s*2)-arc. E

theoren 2.1.6. An (s+2)-arc has no tangents.

Proof: Let A be an (s*2)-arc, and 1et P be a poi.nt on A . Through P pass s+l secants to 4., one for each further

point of A . This accounts for all the lines through P , so there is no Èangent aÈ P . I

Theoren 2.1.7. If s is odd, rn(s) < s+l

Proof: Let A be an (s*2)-arc, ând let P be a point not on

A . The lines through P consisÈ of secants and skew lines, as Èhêorem 2.1.6 ensures no tangents, Thus, the nurber of poínts

on A 1s even, being tl,rice the number of secânts through P . This contradicÈs the fact that s+2 is odd; thus, an (s*2) -arc

cênnot exist, and n(s) < s*l . 2L Gathering these rêsul-ts Ëogether, we have

Theorem 2.1.8. n(s) = s¡1 s odd s*2 s even

Proof: tr'or s odd, theoreos 2.1.2 anð 2.1.7 yielð, n(s) = sa1. Ior s even, theorens 2.1.1 and 2.1.5 yiel-d

n(s) = 5a2. E

Ilenceforth, ra'e use the tern ttovaltt to denote an m(s)-arc in PG(2, s)

Theorem 2.1.9. n(1, s) = s2 + s + I

?roof: EvidenÈly, n(1, s) is the number of points i.n

PG(z, s) E Before proceeding to determj-ne n(k, s) for values of k > 2 , we introduce soue further functions. IeÈ t(k, s)

be the number of ordered sets of k distinct points r^'hich form a k-arc (rnore briefLy, an ordered k-arc).

Lenn¡a 2.1.10. t(k, s) = ¡¡ n(k, s)

Proof: There are n(k, s) k-arcs. Each gives ríse to k! ordered k-arcs, one for each penoutation of the vertices. As these are aLl" distÍnct, and comprise alL ordered k-arcs, we have our resuLt.

Let e(k, s) be the iru¡nber of points whièh inay not be èhosen

to augment ê k-arc into a (k+l)-arc, and let f (k, s) be the number of points r.zhich nay be chosen. Evldentl-y, e(k, s) + f (k, s) = s'+ s + 1 . As we shal-l see, these Dum- 22 bers do not depend on the particular k-arc chosen, provided that k is sufficiently snalJ".

Le¡mra 2.1.11. If f (k, s) is invariant, then t(k+l, s) = t(k, s) x f(k, s) .

Proof: An ordered (k+l) -arc can be obtaÍned uniquely by first selecting an ordered k-arc, and then augmenting this to a (k+1)- arc. The first step can be perforned in t(k, s) ways, and, assrul1ng thåÈ f(k, s) is invariant, each ordered k-arc so chosen may be extended in f (k, s) lrays. Thus t(k+l, s) = t(k, s) x f (k, s) E

n(k' s)--I-f (k' s) cororLaty 2.1.t2. n(k+l, = provided "¡ k+1 , f (k, s) is invariant.

Proof: SubsÈitute the result of lema 2.1.10 for both t(k, s) and t(k+l, s) Ín the equatÍon of l-ema 2.1.11. ïhe resulr is lmediate. E

Theoren 2.1.13. If k < 5 , the quanLity e(k, s) is invariant, and is given by e(0, s) = I e(1, s) = 1 e(2, s) = sa1 e(e' s) = 3s e(4's)=6s-5 e(5, s) = 10s - 20

Proof: Ior = I it is evident that rüe nay choose any point .k , other than the one al-ready taken. No!¡, leÈ the k-arc be A = {?f, ..., Pk}, k> 2 .. 23

We have Èo count the points on Èhe secants of A . There are /r\ \Z) secanEs, each conlaining s+l points. Ilowever, Èhe quan- tity (".1)(5) counts several poinrs mote rhan once. The poínts of A are each counted k-l times, so ne must subtracË

k(k-2) as a correctj-on for this faet. There is also Èhe problen thaÈ, for k > 4 , there are "diagonal" points, that is, points lying on 2 or more secants, but not parÈ of A . It is here that the conditÍon k < 5 appears. Ile noÈe that a point

not in A can lie on' at most 2 secants, for othenTise, k > 6 Now, each diagonal point is determined by 2 secants, t¡hich are in turn determined by any 4 poÍnts of A . Each set of 4 poinÈs of A deternines 3 diagonal points, whence the nunber of dia- gonal poinËs is , (i) . Each diagonal poinr is counred rwice; so !¡e must further subrracL ,(l) as a correcrion. ThÍs yields

e(k, s) = (Ð" . (l) - *,*-', - ,(i) ,

aûd conputíng these quantities for k = 2, 3, 4, 5, gives the

stated results. E

Corotlary 2. I. t4 f(0's)="2*"+1 f(l' s) = s(s+l) f (2, 6') = s2 f(3, s) = (s-1) 2 f(4' s) = (s-2) (s-3) f(5' s) =s2 -ga+2L l,Ie note the consistency of these formulae with earlier theorems.

When s is even, r¿e should have f(s+l, g) = | and f(s+2, s) = g ; for s odd, we should have f(s+l, s) = 0 . These are exacËly the resuLÈs. obtained here for s¡0a11 values of s , naneLy f(3, 2) = I, f(4, 2) = o, f(4, 3) = 0, and f(5, a) = t.

We conclude this section by tabulating f(k, s) and n(\, s) for s = 2, 3, 4, 5, and I < k < n(s)

?able 2.1.1 -- f (k, s)

0 7 13 2L 3t I 6 L2 20 30 2 4 9 L6 25 J t 4 9 16 4 0 0 2 6 5 I I 6 0 0

TabLe 2.L.2 -- n(k, ç)

I 7 13 2l 31 2 2T 78 210 465 5 28 234 Lt20 3875 4 7 234 2520 15500 5 1008 18600 6 168 3100 25

82.2 - Th..e Ovals of PG(z, 4)

We no¡y fix our attention on PG(2, 4) , the finite pJ.ane of 21 poínts and lÍnes. According to Theorem 2.1.8, an oval is a 6-axc, and the nuuber of oval-s Ís 168, as iodicated ln Tabl-e 2.1.2. The resul-ts of Èhis section all corue fron Edge [15i. Several other quantitíes v¡ilI be of value. Iirst, 1et f(k) be the nunber of ovals containing a fixed k-arc. Let gG) be the number of ovals neeÈing a fixed oval exactly ln a specÍfied k-arc. Let h(k) be the n¡¡nber of ovals meetíng a specifi-ed ovaL in exactly k points. Evidently, f, g, and h are defined for 0 < k < 6 The values of f, g, and h are gÍven in

Table 2.2.1

168 48 L2 3 I 1 l0 t2 3 2 0 0 10 72 45 40 0 0

I'Ie nor¿ establish Ëhese values.

Theórern 2.2.1. Any quadrangle (4-arc) is contaÍned in exaetly one oval.

Proof: Rêferr:ing to Table 2.1.1, we see Ëhat f(4, 4> = 2. Thís irnplies that there are only 2 points available for extendi.ng a quadrangle, and so at most one 6-arc. That rre nay use both poÍnts to obtain an oval 1s a consequence of Theoren .2.!.5, or 26

of the facÈ rhar f(5, 4) = l. E

Equivalently, r,r7e may state

CotoTlary 2.2.2. I'f. Z ovals have 4 or ¡nore points in conmon, they are coincidenË.

Eence

Corollary 2.2.3. The vaLues given in Tabl_e 2.2.1 are correct for 4

If r,¡e denote by f'(k, m, s) the number of m_arcs containíng a fixed k-arc (for 0

nT-.r rri - =l T}:.eotem 2.2.4. F(k, n, s) = l:fuff

where we understand that F(k, k, s) = I

Proof: Evidenrly, (rn-k) !F(k, m, s) counrs the number of

ordered sets r{rhÍch nay be used. to extend a k_arc to an m_atc, and this quanrity is exactly fl-l ffi, ,l . E

Qoxolla'ry 2.2.5. The vaLues of f(k) are as asserted in TabLe 2.2.1.

Proof: f(k) = I(k, 6, 4) .

:,JenEoa 2.2.6. h(k) = (l) c(k)

Proof: Each /6\ oval conta:'Lns (n/ k-arcs.

T}reoxem 2.2.'1. The values of C(k) and h(k) given in Table 2.2.7 are cortect. 27

Proof: LeÈ H be a fixed ov4J,, and let A be a 3-arc con_

Èained in II . A is conÈained in f(l) = I ovals. One of

these is Il , and the other tr,r7o have at most 3 points in comon

vrÍth H , by CorolJ-ary 2.2.2. Thus, these tr.ro ovaLs have

exacÈly A in conmon with it , and eG) = Z, No$¿ Let A þe a flxed z-are in ¡I , There êre f(?) - 12 ovals eçnËain-

ing A . Of these 12, one is H Also, there are 4 rÀ7ayâ to choose a Èhi¡d point of ll , and, for each of these cases, there are g(3) = 2 ovals meeÈíng H in 3 poinÈs. Thus g

oval-s of Èhe 1.2 neet E 1n 3 points. Exclud.ing these, we see Èhat EQ) = 12 - 1- I = 3 ", SinilarJ.y, we see thar g(1) = f(1) -'_ (;) c(Ð _ (i) c(r) -48-L-20-L5=t2¡

and c(0) = f(0) - t -h(1) - h(2) - h(3) 168_1_72_45_49 =10

Theoren 2.2.8. In pG(2, s) , an (s*t) -arc has ("lt) "...rr.", (s*l) tansenr", ""u (;) skew lines. An (s+2)-arc n"r' ("ir) secanrs, 0 Èansenrs, ""a (;) skeq i-,lnes. Proof: The nunber of secanrs for a k-arc t" (f) . * Theorems 2.1.3 and 2.f.6, (s+I)-arcs and (st2) -arcs have, respectivel-y, s*l and 0 tangents. IlotTever,

("i') . G+r) = ¿i&r-¿ = (î') so, in both cases, the nr.mber of skew l_i¡.es is 28

62*.*r_s2+¡s+z_/s\'--_ ,-=\,) 8

CoroLJ"ary 2.2.9. An oval- (in pc(2, 4)) has 6 skew lines.

Let us use the tern hÐi1agram for the figure dual to a 6-arc, that is, for 6 lines, no 3 of which are concurrent.

Theoren 2.2.10. The 6 lines skew to an oval form a hexagram,

Proof; Let P be a poinÈ where two. sker¿ lines meet. There are always 3 secants through p (bec.ar¡se there are no tangerit6).

Thus, if 3 skev l-Ínes were concurrent Ln ? , we ehould have

6 lines through P , whereas Èhele are in fact only s*l = 5 such lÍnes. g l,Ie can use the hexagram ske!,r to a fixed oval, A, to define all the ovals dísjoinÈ from A.

Theoren 2.2.11. If the hexagram skew to ¿n oval A be parti- tioned into ÈÌro triangLes, the vertíces of these triangles forn an.oval dlsjoint fron A. Corrversely, every.oval disjoÍnt from

A a¡ises in this way.

Proof: As the 6 l-ines skew to A foïm a hexagran, any parti- tion of the Lines lnÈo trro triangles does, in fact, yieLd 6 distinct vertices, for, otherwise, we shoul-d have four of the Lines concurrent. Let the verÈices of the triangle be pr p?

P' and P4, P5, PU . If sorue 3 of these poÍnts r,rrere colJ-inear, they woul-d have to be divided as 2 f¡on one Èriangle, one from the other. (If all 3 sere fron one,triangLe, they r,roulat, iû 29

fact, be the same point, and 3 of the 6kew lines would then concur in that point).

Assume now thêt Èhe three collinear points are PI, P2, and P4 . In this case, p4 Lies on rhe lines PlP2, P4P5, and POPU, alL of rtrhich are lines of the hexagram. Thus we may conclude that the vertíces of the trlangles forn an ova1, and, as these polnÈs all Lie on l"ines skernr to A , this oval is disjoint from A .

That thÍs accounts for al-l- ovals dlsjoint fron A may be seen by counrins. rhere are t(3) = tt ways of partitioning the hexagram Ínto triangles, and hence, there are 10 ovals disjoÍnt fron A which arj.se out of this construction. The total nurnber of ovals disJoint fron A is e(0) = h(0) = l0 ; hence, aL1 have been construcÈed.

theorem 2.2.L2. Let A, B, and C be three distinct

ovals such that AnB=AnC=Ø Thèn B and c have Lwo poinÈs in connon.

Proof. Let the hexagram of lines skew to A be {a, b, c, d, e, f] . Then B and C arÍse from partiËions of the hexagram into triangles. Let these parÈitÍons be, respectlvêly, abc/ d,ef. and abd/cef , It is imnedLately evident that Ehe poitrtE êb and. ef are eon¡non to both B

and C , and additional conmon polnÈs wouJ-d violate Èhe hexagran property. 30 Corollary 2.2,13, There do noÈ exÍst Èhree nutual_ly disjoint

ovals .

We reca11 that, by Theoren 2.2.1, art oval is deternined by arry 4 of its points: This fact permits us to describe a geoneÈric method of finding the 2 extending poinËs, when given â 4-arc.

Theorem 2.2.14. If A is a quadrangle (4_arc) , the diagonal points of A are coLlinear. This line contains 5 points. The two points which are noÈ diagonêl poÍnts

are the points lrhich extend A, Ëo an oval-.

Proof: that the diagonal poínts are coLlinear fol-l_ows fron Ëhe fact that s = 4 is a power of 2. The nunber 5 , of poinÈs on thÍs J-ine, Ís, of course, s*] . Removing the 3

diagonal points leaves exactly 2 poinÊs renainíng. LeË Êhe points of the 4-arc be pr, ..., p4, and let the trüo construcÈed points be X and y . That no 3 of theêe six

poÍnÊs are collinear fol-Lows by eonsidering tlro câ.ses. ¡'irs!, Ìr7e can noË have ?rr X, and y collinear, for thÍs vould mean thaÊ p. is on the line of diagonal

points, and, hence, infact, is a diaþonal point_ Second.ly, we can not have P-.p- , and one of X or y coLlinear, r-- J for P,.P.rJ meeta rhe line of diagoheí pointê in one of the

diagonal poi.flts, not in X or )f . Thus, {Pr, ...' P4' x, Y} Ís a 6-arc or oval¡ by Theoreo 2,2.1, ÈhÍs is the irnique ova1 containÍng

PL, P2, P4 . "3, 31 theoren 2.?.15. LeÈ A, B, C. be ovals Êuch rhar len ¡l = Z and llncl =0. rhen lBncl =g ., z.

Proof: Ler A¡B;{p,Q} . Then A;{p, Q} +SO, where qO is a quadrangle. The line pQ contains three.other pointe,

Dl, D2, and D, , which are, by Theorem 2.2.14, :che

diagonal poÍnts of qO NorÀr, the sides of q0 passíng through D, conÈain 4 points not associated r,,rith q0 . These forn a -quadrangle ql . IË is clear that A1 = 9l + {P,Qi ís a 6-arc. If 3 poinËs of A, are collinear, they can not aLL come from q, (which ís a _arc), nor can they be p, Q and one frorn q, (e1se the 1atÈer point would be one of DI, D2, Ð3). This leaves rhe possibllÍty p Èhat the poinËs are and ÈÌro points of q, . The ûro from ql ian not Lie on a side of qO through D, r because thÍs l-ine does p not meet . Thus, we have ql = {Xt,yl,X2ry2 } where XrX, and yry, meet in O, , "r. sides of q0 , and the points Xl, Y2, and p are coll-inear. By Theoren 2.1.6, XL[2P is a secanË of A , ând meeËs A in another poínÈ, R . R .can not be Q, ; so R is one of the poinÈs of q0 . Hence RDlXlY2 ts a side of both qO and {1 r and contâíns

ânother poÍnt of qO ; thus Ít contêins 3 points of A _ a contrad.iction. CoÊs.equgntly, A1 i.s an oval, as l_ikel¡ise are the sets A2 aûd A3 determineai analogously fro¡l the diagonaL points D, . and Ð, via quadrangles q2 and q3

It is clear ËhaÈ the quadrangles qi are mutually disjoint (and extraust the 16 poinÈs of ou: pJ.ane àot on pQ). 32

Since g(2). = 3 , Ar, Ar, and A, are èxàct1y the 3 ovals mêeting A Ín {P,Q} .U"o"" B is one of A' A2, A3 -

Consi.der nor^¡ the 5 lines through Dl . One is PQ , and ttro more are comon sídes of q0 and ql . Eence, the other Èlío lines are comon sides of q2 and q, . Likewise, con¡on sides of. 96, g2 and 91 r {3 pass through D2 . tr'or

D3 , the pairing is q0, q3 and qr, {, .

Non consider the hexagram of lines skew to A . These 6 l-ines each neeÈ ?Q , but not 1n either of the poi.nts P or Q . Thus, they nust pass, in pairs, through each of

Dl, D2, and D, . Ttre tr{'o lines through D, do not meeÈ qO and thus can not neet ql Thus, they üust be the lines foruring sides of q2 and q3 Sinilar resuLts hold for the lines through Ð, and D, .

By Theorern 2.2.11, C is ao oval obtained by partitio- nlng Èhe hexagram into Èriangles. There are two r4rays i-n which this can happen. In the one case, the sides of one tríangle cotÈain each of Dl, D2, and D3 . Then the sides of the other triangl-e do Liker¿ise. Conslder the vertex on the lines through D, and D, . The line through Dl contains points fron q, and q, . The 1Íne through D2 contains points fron q, and q, . Thus, this vertex lies in g, . BuÈ this happens for both triangles similar resulÈs occur for D, and

D, and f.or D, and D3 . Hence C meets thTo points fron each of ql, q2, and q, , and hence fron Arr Àr, and A3 . 33 But one of these is B , so ln n Cl = O . (This accounts for

4 of the possible ways of fixÍng C disjolnr frop . A.)

In the other case (which accounts for the other 6 choices for C), the first triangLe has ttro sides meeLÍng in,

say Dl , and one passi.ng Ëhrough, say D, . The second

triangle then has Etro sides through D2 and one through D3 . tr'rom the first Ëriangle, r,¡e get Dl and ttro points of 9,2 . Froo the second, \ùe get D2 and tl¡o polnts of ql . Thus lcnorl =lcnorl =z and lc o orl = o . Again, B is one of Ar, A2, . This accounts for all ^3 the cases. E

CoroLlary 2.2.16. If lA n Ol = 2 , there are exacrl-y rtÍo ovals c such that ll n Cl = lr n Cl = O .

Proof: Usifig the notation of Theorem 2.2.15, rùe rûay assune

B = At . C must be obtained by partitionÍng the hexagra.E of Lines skew Èo A so that the lines through D3 are separated, and the lines Èhrough Dl, D2, respectiveLy, not sepêrated. lhis can happen in only 2 ways. E

52.3 - The Block Design (56,11,2)

We nonr see how a (56,11,2) block design nay be obtaíned froo the geometric properÈies of oval-s in pG(2,4)

The results of Èhís secÈion are due to Èhe auÈhor.

IrIe now fix oul atÈention on any one oval, A .

Let a = {xl ln n xl = O (2) , and X. is an ovai } ' 3r4 Theorem 2.3.1. lal = 56

Proof: Evidenrly lal = rtCO¡ + h(2) + h(4) + h(6) =10+45+0+t =56

Theorem 2.3.2. If Xea aad Xny=@ then yea. Proof: there are clearly 4 cases depending on lX n nl I'irst, if lXnAl =6, X=A and the reaulr is rrivial. Second, if n e,l = 2 ,"¿ lX lX n:Vl = 0, rhen je n yl = O or 2 by Theorem , 2.Z,IS. In eÍther câse, y e A . rhirdly, if lxnel =0 and.lxnvl =0, rhen jvnel =z by Theoren 2.2.L2. trinally, lX n al = + is not possible, by CoroLLary 2.2 .2. E Now, d.efÍne a reLation = on A by X=y Íf lxnrl =0,or X=y. Ler [X] = {vlx = r}

Lemna 2.3.3. The reLation = is synmetrj.c, and so ls the incídence n¿trix for = . rheoren 2.3.{. ltx:l = 11

Proof: ltx:l = h(0) + I =10+l

=11 E rheoièn 2.3.5. Ifx,yeaandX#y, llxl n lyll-2. Proof: There are ttüo cases, lX n tl = g or 2 .. If lxntl =0, rhen {x,y}ctxlntyl 35

By Corollary 2.2.L3, there are no other ovals disjoi.nt from X and Y;

so lfxt n fY:l = lIx,x]l = 2 . If lX n fl = 2 , there are exacrly two ovals disjoinr froû both, antl rhese comprise llXl n lyJl

Corollary 2.3.6. The incidence maÈrix, M , for = , yiel¿ts a s)¡meÈric balanced incompLete bJ_ock design r¡hich has parameters v=b=56, r=k=11 , L=2. The design is self-dua1.

Proof: v = b = 56 expresses the size of M , and hence the dinensions of the incidence natrix.

Now, r = k = 11 is the resul-t that each bl-ock has l1 elements, together with the fact that M i.s sy metric. Fínally, \= 2 is obtained frorn Theoren 2.3.5, again using Èhe syrmeÈry of M , I 36

Chapter 3 - T¡^ro Cubic Surface Block Designs

In this chapter, rüe present the structural geonetry of the doubLe-six configuration, and indicate its relation to lines on a cubic surface in 3-space. I'rom this, qre derive

s)rmmetrLc balanced incouplete block designs r,ri th parameÈers

(36, 15, 6) and (45, 12, 3') . tr'or this chaptet, v¡e confÍne our attention to

PG(3, R) , real projectÍve 3-space. Before proceeding with a discussÍon of the double-six, we sunnarlze properties of ruled quadrÍc surfaces, and consider related topics. The theory in the next section is presenÈed wÍthout proofs. (cf. Baker [3].)

53.1 - Prelininary Topies

A ruled quadric surface ls a set of poínÈs whose homogeneous co-ordlnates, after proper choice of axes have been nade, satisfy *l**2=x3+x42222 ln linedr algebraic terms, a ruled quadric surface is Ëhe set of points determined by

Q!*r, "2, x3, x4) = 0 , 37 where Q is a quadratic form of rank 4 and signature 0 the properties of ruled quadric surfaces are enumerated in the folJ"owing sequence of theorems.

Theoren 3.1.1. Through every poinË on the surface, there pass exactly two (dÍstinct) lines lying wholly ra?ithin the surface.

Theorem 3.1.2. The lines of ruling may be partitioned into two fanilies tríth the property thaÈ two lines from the same famil-y never meet, but tr,ro lines from disÈlnct farnÍlies always meet. Consequently, Èhe lines of ruling through a fixed point comprise one fron each of the Èwo fauil_ies.

Theoren 3.1.3. If two distfnct quadric surfaces have two Lntersectíng lines as comon poifrts, then their remaining common points 1ie Ín a single p1ane.

Theorern 3.1.4. A ruled quadric surface Ís completely deÈernined when three distincÈ lÍnes fron one of its fanilies have been given.

[Ie shal1 return to Èhe ldeas of theorem 3.1.4 shortly.

A set X of l-ines is called sker¡ if no Ëhro of Èhe lines meeË.

A line I is called a transversal of a seÈ of lines X if f meets each member of X . With this terminol-ogy, r{re can re- phrase Theorem 3.I.2 as The lines on a ruled quadric surface are partitioned ínto tkro sker'r sets. Each line of one set is a transversal of the other set. 38

tle nor¿ surma::ize. the nature of transverbals for

sß,aJ-1 finiÈe sets of lines.

?heoren 3.1.5. A set of ú,ro skêw lines has a uníque transver- sa1 through every point not on one of the 1Ínes.

Theoren 3,1.6. A set of three skev¡ lines has a unique trans- versal through any point on one of the Iines.

According to Theorem 3.I.4, a set of three skew

lines is contaíned in at most one ruled quadric surface. I{e

nor see that such a seÈ is, in fact, contained ín exactly one ruled quadric surface.

Theore¡n 3.1.7. The set of points on the famÍly of transversals to a seÈ of three skew lines forms å ruled quadrJ-c surface. The tra¡sversals are one of the skew sets of lines on the sur-

face, and the Lhree given lines are in Èhe other skew set.

Theoren 3.1.8. A set of four sker,¡ lines having one txansversal has, in general-, exactl-y one ¡¡ore.

Proof: tr{e give the proof of this theoreD nainly to indicate condÍtions under r¿hich it fails to ho1d. Let Èhtee of the four Iines be fixed. They determine a unique rul-ed quadric surface, and lie in one of the skew fanilies. The transversal 1i-es in the second f aniJ-y. lhere are several possibiLities for the fourth 1ine. ¡'irst, it nay 1ie on the surface, as pårt of the first skerar fard-ly. In this case, the four have an infÍnÍtude of transversals (a11 of the second fanily). Second, it nay be 39

a tangent. In this case, there is no further transversal_.

Finally, if neither of the first tr¡ro cases holds, the fourÈh line r¿'il1 meet the surface in trüo distinct points, one of rzhích is already knom, namely, its poiût of inrtersection with

the given ttansversal. I'ron Theorems 3.l.L and 3.1.2, we see

that there is a l"ine of Èhe second skew family through the other poínt of intersection r¿Íth the surface. This is, then,

a second connon ttansversal to the set of four skew lines, and no further Èransversal 1s possible unless the four Lines 1ie whoJ-ly ín the quadric surface deternined by any three of then. I

53.2 - The Double Six Configurarion

The theorems in this section are Èo be found in Baker [3] and Henderson [2] . The notation for the double- sÍx is due to Schläf1i [31] , and âdditional hÍsËorical oaterial nay be found ín Sylvester [39].

A double-six or Schläfli double-síx fs ê set of tr.¡elve distinct lines {a1, bl, b2, b3, b4, b6} ^2, ^3, ^4, "5, "6, b5, sdth the intersecÈion properties 1) ê. meers Oj for i f j 2) Èhere are no other intersections 3) no Èhree of the l-Ínes concur.

Lê¡jomâ 3.2.L. A seÈ of 12 lines satisfying properries 1) and 3) above. ¿nd havj.ng one other Íntersection, lies in a plane, in which case all U-nes meet; or in a ruled quadric, with the a. lines in one skew farnily, and the b. i.n the other. 40

Proof: There are Èwo cases to consider, namely the case that

al meets a2, ox that .al meets .b, Id the fi.rst case,

a'rd. a, define a plane r . ALl Ëhe lines "1 b3, b4, b5, b6 lle in this plane, as they meet and "l a2 . Then, .3,

and aU lie in thi.s plane, meeÈs ^4' ^5, as each three of the b-lines. Fínally, thís places bt and b, in the plane. In the second case, consider the ruled quadric sur- face generated by a¡, a2t a, . Let A be the farnily of

ruJ-ings including a3 . Let B be f aa-ily. ^I, "2, the other Now, bl, b4, b5, b6 are transversals of ay a2, a3 . Thus bl, b4, b' bU e B. tr'i.na11y, bZ aad b, e B ; so all of

Èhe lines lie on a ruled quadric. 8¡

It is by no meaûs obvious Ëhat double-sÍx confígura- tíons actualLy exist. This ís guaranteed by the fol,lowing theorem, which nigh appropriately be Ëerned the ,rl'undamenÈal Theorem of Double-SÍxesr'.

Theoren 3.2.2. Let l-Ínes a1r .,.r a5 and bU be glven, so

that b6 is a transversal of al, ..., a5 . If the l_ines al, ..., a5 are placed so that each four of then determine uniquely another cotrmon trânsversal (as Ín Theoren 3.1.8), then there is a unique double-six incJ-uding the gÍven lines. Proof: There unique are J-ínes .bl, bZ, b:, b¿, and b, such that each meeÈs (alJ- four of ."1, ..., a5 except.1ts like- nunbered nate). It remains to be seen that bl, b2, b3, b4, and b, have a conunon transversal. Any four of the latter 4T

have a transversal noÈ.yet noted. For example, bl, b2, b3, and bO all are.met by .5 and a unique other 1ine, ,. .

I-ikewise, bl, b2, b3, and b, have aO and another l-ine n as transversals. If ¡,re can shor,r that & and n coincide, then, denoÈÍng this sLngle 1ine as a6 , we have cons.tructed. a coroplete double-six in a unique r^ray. AlternatÍvely, ve na'y .is show that .0 compLetely determined by the lines uL, "2, ê3' b1, b2, b3, and bU . We shoul-d then conclude thar !, and m coinclde.

Consider rhe nine points pL2, pzL, pt3, pZt, pZ3, PlZ, Pt6, PZ6, P36, where pr. Ís the inreïsection of .i r'ri th bJ . Now, conslder the t¡üo quadric surfaces generated bY aT ar, a, and brr bZ, b4. Theae two eurfaceg.share lines and bO (whÍch "5 ureet one another) and so, by Theorem 3.1.3, Lhefr remaining connon points J-íe in a single plane. ,23 and. Py are such cormon. points, ês is pt6 . These three points thus deternine the plane. I'inalLy, l-et X be Ëhe intersection of b, and & X ís colmon to both " . quadrics, and so 1iés in the plane of pr', p23, p32, where

this plane is xoet by bl . The sane argument vilJ- place the intersection of b, and m in this plane, again at the poÍnt X . ?hus, L, n and b, concur ln a single point, X . Likenise, Z, n, and b, concur in a poinr Z (in plåne p36, ,I2, ,2L). Thus, Z.and ¡n are identicaL, and yield the fínaL lLnê, a6, for Êhe double-sfx configuratfon. gl 42

Having been gÍven a double-six eonfigurati-on, we may

derive further Lines from it. tr'or exànp1e, since al neets b2, they define a plane nrr. Likewise, a, and b, define another plane nrr. The intersection of these two planes is a line ¡shich we may denote either by o" .21. This nethod "L2 al1or,¡s us to derive = tt lines c-., . I,Ie stare nor^r, but reserve 11)l't"l Íl for Later proof, a fundamental theorem.

Theore¡n 3.2.3 The 27-1ine configuration of 12 lines in a double-sÍx and its 15 derlved lLnes forn a closed system,

ín the sense that, finding a second doubl-e-six amongst the 27 llnes, and deriving 15 ner¿ lines fron thls, we are 1ed to the same total-ity of 27 lines as before.

Ite cal-1 Èhis configuratÍon a complete double-six. It 1s connected r,rith the set of lines on a general cubic surface;

we indÍcate this connectÍon by statÍng r,rithout proof several_

theorems .

Theoren 3.2.4. A general cubic surface conÈains exactly thTenty-seven li-nes.

Theoren 3.2.5. The tr,renÈy-seven 1Ínes on a cubic surface forn a complete double-six.

Theorem 3.2.6. There is exactly one cubic surface containing a conplete double-sLx. 43 Thus we see that the study of'the conplete double_sÍx Ís exactly the study of lines upon a general cubic surface. ïÍhi1e ¡¡e have not proved thi.s connection, we shaLl- feel free,' to use terninoLogy which arisês naturê]-1y therefrorn. our lesults do not in any Íray depend on Èhis connection vith cubic surfaces .

53.3 - The StrucÈure of the Conplete DoubLe_Six

The theorens 3.3.1 - 3.3.10 which establ_ish the structure of the complete double_six come from Baker[3], and Ilenderson [20].- The conbinatårial results 3.3.fI _ 3.3.L4, whereby the (36, 15, 6) design is obrained, are due to Ëhe author.

Theoren 3.3.1. The 27 lines of a conplete double-six have the folJ_owi4g Íntersection propertLes: l) a-aJ neers b. if r + J; 2). a.i meets____ c.j = cji; 3) b._j ____"neets .ij = "jii , 4) ci¡ neets cür Íf {i,'j} n 1r, n} = Øi 5) there are no further intersecÈions. Proof: The first property folior^rs from the definition of Èhê doubl"e*six, and the second and third properËies fol-1os dÍrectly from Ëhê definiÈion of the dêrÍved 1lnes c... t[e urílize rhe proof of Theorem 3.2.2 r" :;." the dual of property (4). es Ín that rheorem, ler p,. be r-J the point of interaection af the Lines a. and br. 44 Now, leÈ urj = UJa be the line joining trj ao tjr. Noring Èhat we inay identi.fy rhe poinr X of Theoren 3.2.2 as p;r,

¡ve have that Pl6, P6t, PZ3, and P' are alL coplanar. Thus,

dr6 neets drr. The dual resul"t is that cl6 meeÈs c23. Ttlís case is general.

FinalJ-y, r^re sho¡¡ Èhat no further intersections

can occur. There are t!ùo cases to consider. We may have a derived line meeting a line of the origÍnal double-six, or we nay have È¡,7o derived lines neeting. As an example of the

first case, suppose that al meets c23. Then al, b, and c* all- neet, and hence l-ie Ín the pJ-ane nrr. LÍkewise, the lines "1, b3, and c, al-l meet, and thus 1ie ín pLane nrr. These ü'ro planes, conÈaining both al and crr, must coincide; thus, b, rneets b, and we have viol,ated a conditioû on the original double-six.

ln the second case, suppose thaÈ cl2 ûeets cl3. Now, b, and b, meet both these lines, and consequenÈly lie in the plane determined bV cu and crr. Thls means that b2 and b3 meet, ¿gain a violatÍon of Ëhe double-six requireÐenÈs. I This theoren is the foundati.on upon ¡,,hich all of our structurâ.l results rest; Ëaken as a sÈarting poínÈ, .it.leads to the block desigas nlthout regard to the reêlity of the complete double-sÍx. 45 Theoreo 3.3.2. Every line meets exactly 10 others.

Proof: Lines al and c12 are general cases. Line al meets bz b3, b4, b5, b6, and crr,. .14, .15, and no "13, .16, others. Line c* meets aI, b7, br, and ca6, c4S, c46, "2, "34, "35, c55,r and no others. E

Theore¡n 3.3.3. There axe 276 paÍrs of skew lines, and 135

pairs of lines nhi"ch neeÈ.

Proof: Each l-ine neets 10 others, and is skew !o 16. Thus,

the number of ordered skew pairs is 27 x 16, and Èhe number of unordered pairs 1s 4 x 27 x 16 = 2L6. Sirnilarly, the nuuber of unordered pairs of lines which meeÈ is r¿ x 27 x 10 = 135. I

Theorem 3.3.4. A pair of sker¿ lines has exactly 5 transversals.

Proof: We consider cases. Lines a, and a, are skew, and meet each of br,,b4, b5, b6, artd crr. Línes a, a:nd b, are skev, artd meet each of .r4, Lines al and c* are skev, "12, "13, "15, "16. and meet b2, b3, cl4, cl5, cl6. Finally, J-ines cu and cß are skew and meet al, bl, .46, T'l'ìis a'ccounts for all "45, "56. possible eases. E¡

In our original double-six, rüe say that a. and b. are nated. It is clear that the property of beÍng nated Ís intrÍnsic to the double-si.x, and is not deperident on the r,ray in r¡hich we assign names to Èhe E!,relve lines. LeË us use the notation 46 ("" u2 t3 .s t-l "4 "ul lot bz bs b+ bs ouj

to asse?t that the set {ar, ..., ã6, b1, ..., b6} forrns a double-six.

Theorem 3.3,5. The::e ar,e at least 36 double-sixes.

P::oof : lle provide a list. There is the original- double-six.

Now f":.f t46 t-l ^2 "g "56 "uuJ t13 b+ bs ou ["r. "!2 ,|

is a double-six, and. ís ct¡aracte::is ti c of l'?l = ra examples. 1.3 j Likewise l. bt l ltt "2a "2+ "2s "rul tt bz t13 .15 .tuj [t, "14

is a d.ouble-six, characteristic of = tu ..""". [3J Thus, at least 36 = ! + 20 + 15 distinct d.ouble-sixes exist. I

Theorem 3.3.6, A given pair of skew lines can occur as nates in at most one double-six.

Pr.oof: Given two lines, we ::equire ten more. There alîe ten

Iines meeting the first given Iine, five of which also neet the second. These five cannot be used. Likewise, of the ten lines meeting the second, fíve also meet the first, and so cannot be used. This gives us exactly ten lines f:rom which to choose ten, so hte can construct the double-six in at most one r^ray. E 47

Co¡o1l-ary 3.3.7. There ar?e exactly 36 double-sixes.

P¡oof: lte have 216 pai::s of skew lines. Each can occur as nates in at most one double-six, and each doubl-e-six accor¡nts fo:: 6 mated pairs. Since 216 = 36 x 6¡ we have accounted fol? all the pairs, so no funther double-sixes a?e possible. E

Theo¡em 3.3.8. For a given pain of lines $rhi ch meet, there is exactly one further ]-ine meeting both of the given lines.

Proof: We ill-ustrate by cases. Lines aa and b2 meet c12. Lines aa *d .1, meet line br. Lines c, and crU meet line cr..

These account fon alJ. possible cases, E

Sets of three lines, each two meeting the thind, are calLed tritangent planes. This teminolos/ a::ises f:rom the r€lation wíth cubic sur.faces in an obvious way.

Corol"lary 3,3.9. Ther€ a?e exactly.45 tritangent planes.

Proof: The:re are 135 pairs of íntersecting lines,

Each tr:itangent plane contains 3 of these pai:rs, so the nunber of trita¡ìgent planes ís 1/3 x 135 = 45. E l,le a¡e now in a position to prove

Theoren 3.2.3. The conplete double-six is a closed figu::e, in the sense that ftrorn any double-six amongst the 27 lines, the 15 derived lines ar.e the ::emaining 15 l-ines of the figur.e.

Proof: We show the tr4lth of this theorem fol: the two double- sixes of theorem 3.3.5. Having picked a double-six, we will

¡ename its lines Arr ..., AUBI: ..., B6 and .Let C.. be the 15 48 der"ived lines. I{e have to show that the de::ived lines ar:e lines of the or'íginaI configr:ration.

In the first case, let

A1 =t1 B! = "23 A2 -- ^2 -2 '13 t3-o3 83 = t12

"4^ -^- "56 Br, = b+

"5 '46 B-55 = b- A6 Bo=bs = ",+5

The ::esults we requine car¡ then be obtained f::om Theor"e¡n 3.3.7.

Fon example, Cr, lies in the plane formed by .1 .rd ClZ "13. also lies in the plane formed by a, and crr. Now b, is the unique J.ine common to both planes. Thusr Cr, = br. Similarly,

Clg b2' C23 b1, and Cuu C46 a5r Cru au. Now = = = "6, = = CrU ìies in the plane deter"mined. by a, and bU. CrU also lies in the pJ"ane deterrnined by cUU and cra. The single line common to both planes is crU. Thusr CrU = c14, and simi1ar"1y, C_.., j ll = c_..al for" i = !. 2. 3 and = 4, 5, 6.

In the second case, let

A1 = B!= "1 ^2 AZ=br Bz=bz A3 = 83 = "23 "13 A,+ = tz+ 84 = "14 A5 = "25 85 = t15 A6 = "26 86 = t16 49 ft is immediate that C!2 Nowr Cra lies the pl_ane = "!2. in fo::med by *d and also in the plane fo::med by a, and "1 "1A, "2g, Thus Cra = br. Sinritarly, Ct,* = b+, C15 = b5, C16 = b6, and Cr, =.3, C2+ = C2S = aU, CrU = aU. Finally, C34 = ^4, "56, C35 = C36 C45 C46 and Cru cru. E "46' =.4s, = "36, = "35, =

For the doub.le-six t3 t4 ts ("" ^2 "ul bz bg b,* bs [0, ".-J we say that {ar, ..., aU} and {b1, ..., b6} are the t:ows of the doubl-e-six. It is evident that the partition of the twelve lines into two rows is deter:mined nithout reference to the 1abe11ing of the Lines.

Theorem 3.3. 10. Two distinct doubl-e-sixes are r€Iated ín one of tr^ro ways :

í) each ::ow of the fir.st shares one line element with each :row of the second, o:: ii) one row of the first shates thr"ee lines v¡ith a ¡ow of the second, and sha::es no lines erith the other; the second rovr of

the first shares r:espectively 0 and 3 lines with the rows of the second.

P:roof : We rìay take one of the double-sixes to be

[^" ^2 "3 t+ "s t.] þt bz bg b,* bs orj and the other to be f"l. bt "2g "24 "2s trul l, bz '13 '14 '1s 1.,| oll ¡. I rll"r ^2 "3 "56 "46 "+s I b,* bs ou ["r. "1a "!2 ,J

The r€su1ts follow by inspection, E

Nor^r, let I be the set of doubl-e-sixes. Ðefine a

r€latíon, = , on á by X = Y if and only if the ror^¡s of X and Y share one line each.

Leûna 3.3.11. The ¡elation = is imeflexive and symmetric.

Theo¡em 3.3.12. ff X = Y, there are exactl-y 6 double-sixes, Z, such that X=Z and Y=2.

P::oof: lle may suppose that .s *= l'. ^2 '3 '4 "4 þt bz bg b+ bs bJ and ("" bt "2g "24 "2s "r.l v = l- o2 t1t+ t15 I . ["t "1g "tuj

Let (i, i, k, ß, rn, n) be a pexmutation of (1, ..., 6), Then the most genenal doubl-e-six related to X is bi tjr tjr,] |r"t "jr "j, D..rl = I oj tik ti"JI t": "it "i, and we see innediatel-y that 51 Y D_., and = r-l if only if 3

Theorem 3.3.13. If Xf yandX I y, there are exactly 6 Z suclr that X=Zr and y=2,

P¡oof: We may assume

[t ^2 '3 '4 "s -oJ X= I I lot bz bs b,n bs ouj (^, tg t56 t46 ^z "urJ v= I r b,* bs ou l"ru "13 "!2 J

Letting Orj be the same as in the pr:evious theorem, we find that if and onty if 1

Theo:rem 3.3.14. The incidence mat::ix fo:r = yields a (36, 15, 6) block design. ''', Pr-oof: Since the¡e are 36 double-sixes, v = b = 36. Ile have noted that the ¡elation = is symrnetr.ic. With the count given in Theoren 3,3.5, we see that:r = k = 15.

Finally, Theo¡ems 3.3.12 and 3.3.Íì together impty À = 6. B

93.4 - The Trltarigent planes In rhLs section, we develop Èhe (45, 12, 3) design frou 5e

the 45 tritangent pLanes. Theorems 3.4.1 - 3.4.4 cone from

Baker [iJ and Ilenderson [20]. Theorems 3.4.5 and 3.4.6 are due to the author. Theor.em 3.4.1. Every line l-ies in five tj?itangent planes.

Prrcof: According to Theorem 3.3.2, a fixed line meets 10 othen lines. Then, by Theo¡em 3.3.8¡ they must pair" off, to

yíeJ.d 5 planes. E

Theorem 3.4.2. Each tï'ita¡gent plane shar:es one Line with 12 other?s.

P::oof: A given tritangent plane is couposed of three Iines.

Each line Lies in five planes, one of which is the plane under

consider"ation. Thus, the total numbe:: of planes shaning one l-ine with the fixed plane is 3 * (S - lJ = !2. [i

Now, let .4 be the set of tl?itangent planes, and define a rela- tion=ron¿by

X = Y if and only if X and Y share one line .

Theorem 3.4.3, If X = Y, then the¡e a¡e exactly 3 tritangent planes Z such that X=Z and Y =2.

Pnoof: Let 0 be the l-ine shared by X and y. The::e ar"e five planes ineluding L, by Theo::ern 3.4.1. Two of these are X and. y.

The other three serve as choices for: Z. Novr suppose that X = {t, .!, .2}, Y = {!,, 11, r2}, and Z = {mr, n' p}. Then f,r mrr and n, meet in pait:s, and so they fo¡m a tr:itangent pLane. This means that X = Y. Thus, the only choices for Z a::e 53 the 3 additional planes through. 4.

Theorem 3.4,4. Let X = tLf, t2, Lgl and Y = {!!1, *2, rg} be two planes with no line in common. Then, by renumbe::ing if necessari¡, we may assert that li meets rj if and only if i = j.

P::oof: We rnay choose X b2, The = {a1, "I2}. proof follows by cases. For example, if Y = {ar, ba, crr} , then

a, meets brr b2 meets , *d "23 c12 neets a2 .

If Y = {crr, crU, cr.} ,

al meets c14,

b2 neets c23,

c12 meets cS6.

If Y = {a3, bU, caU} ,

a, neets bU, b2 meets a3,

c12 meets ca4,

Al-1 othe¡ cases foJ.low similanly.

Theorem 3.11.5. If X I Y and X / Y, then there aÌe exactly th::ee Z such that X = Z artd Y = Z. g,2, g.gl Proof: Let X = {L!, and. Y = {mr, nr, mr} whe::e .Q,. ¡T€ets m.ItJ- . with no othe¡ inter:sections. Now .{,, and n. neet. 54 and thus form a tritangent plane with some furthe:: line n.. This gives 3 planes Z. No other possibilities exist, fon othe:rwise we should have .1,. meeting m. for" i / j. E

(+S, Theore¡n 3.4.6. The incidence matnix for: = yields ¿ 12, 3) bloc)< design.

Pnoof: v = b = 45, because of the nunber of tritangent planes.

Noting that = is symnetri c, l¡e see that Theor:em 3,4.2 gives î=k=!2. Finally, À = 3is a consequence of Theo::ems 3.4.3 and 3.4.5. m 55

Chapter 4 - Isomorphisn of Designs

In this chapter, ¡ve díscuss tr.ro general ways of

testing whether designs wiÈh Èhe same parameters are isomorphic. Both rnethods of testing r{ere conceived by the author, although l¡al-lis [44] and Ner,¡man l28l have discussed Ëhê first nerhod as an isomorphisn test as we1l.

54.1 - The Snith Norrnal ¡'orm In this section, theorems 4.1.f - 4.1.16 3re taken fron Sranton 1371. Theorerns 4.1.17 - 4.1.23 axe fron Wa11is [44]. l{e recall that we have used the term incidence matrix in Èhe follor¿ing way: given a balanced incomplete block design

¡r-Ith paÌameters (v, b, r, k, À), let. the varietíes be {Vl, V2, ..., Ur} and let the blocks be {Br,82, ..., Bb}, t¡here each seÈ is numbered Ín arbitrary fashion. Then define the vxb måtrix A = (a..) by

aJ 0 if v.r-J I8.. The baslc properties of A are

(í) the colun¡r sum6 of A are a1L k;

(ii) tfre rnarrix ÀAT has r on the diagonal- and l. elsewhere.

Conversely, if a matrix A sati.sfies conditions (i) and 56

(ii) above is given, then the reverse procedure yields a bal-anced incomplete bJ-ock design with par.anete::s (v, b, r, k, l).

Before proceeding with ou:: discussion of block

designs, we conside:r some prope::ties of permutation matnices.

A perrnutation mat::ix ís a squa::e matrix consisting of Ots and

1rs such that each ::ow and each colunn contains exactly one 1. The basic theoren on permutation matrices is

Theo:ren 4,1.1. The set of n x n permutation mat¡ices forms a group isomorphic to Sr, (the symmetnic gtoup on n synbols).

P::oof: Let c be a pernutatíon fr"om Sr, . Define the n x n (a..) natrix A= rl bv (! ir j=a(i) a..-ij = I )(o ir jlc(i)

It is clea:: that A is a permutation mat:rix, and that the r€fa-

tion c -> A is invertibfe. We need only show that matnix

multiplication is equivalent to group nultíplication in Sr, . To this end, .Let ß be anothen permutation in Sr, , and B (b-.") = .rl be the cor:r:esponding matr"ix. Now .'6 = dÊ means ô(i) = 8(c(i)). thus

... It a.. b- . .tl = -t=t .rr( lcl-

I'le know that uik = 0 unless k = o(i) , and then .ik = 1 Thus , bo(i).j. "ij =

By the - definition of b..,r-l 57 1 if j = ß(c¿(i)) = ô(i) rl , o if i/ß(c(i))=ô(i)

Thus C is the natnix colîlesponding to ô = aÊ . Corolfarl¡ 4.1.2. A-1 = AT foo perrnutation natrices.

ConolLar5¡ 4.1.8. lf A is a permutation matrix, then det A = 11. Proof: Let det A = a. I,Ie have I = AA-1 = AÁT, and thus 1=detf = detAdetAT= ¿2 Hence a=t1 . E

A trEtrix A with integer entries and deteminant t1 will be caL.Ied. an integer non_singular natrix, o:: fNS:mat¡,ix. Thus we obtain

ConoLlarS¡ 4.1.4, pe¡mutation rnat¡ices a¡e lNS_rnatr:ices.

The utí]ity of permutation natrices in connection with btock designs a::ises fi:om the welL_knor+n

Lemma 4.1.5. If p is a permutation matnix, and if A is any matnix, then pA is a nat¡Íx which can be obtained. frorn A by row permutations. conversely, any ¡ow pennutation of A can be t€alized in this way, and sueh cor.r:espondence is biunique.

A simila¡ result hol"ds for cofurnn permutations, but the permutation. natrix appea¡s on the right.

Let A and B be two matnÍces of the same size. we vir'l say that A is pe:rmutation isomorr¡hic to B if B can be 58 obtained from A by use of row and colu¡nn penmutations. In view of LeTffna 4.1.5, this is equivalent to asset:ting that permutation mat¡ices P and Q (of app:rop::iate sizes) exist, such that

B = PAQ.

It is easy to check the following result.

Lemna 4.1.6. Permutation isomorphism is an equivalence relation.

Lemma 4.1.7. If A is the incidence matr"ix for" a balanced incomplete block design with parameter.s (v, b, n, k, ),), and íf B is permutation-isomorphic to A, then B is also the incidence mabrix fo:: a balanced íncomplete block design with parameters (v, b, n, k, À).

lle note that two balanced ineomplete block designs Ïríth paraneters (v, b, r, k, l.) are isomorphic if their" inci- dence matri ces are permutation-isomonphic.

A canonical- form for: permutation-isomorphisrn is not known. As a result, al,gorithms for testing pe::rnutation- ísomorphism of Tnatrices generally take the form of merely testing llow and colun¡rì permutations of one matríx against the othen. However, there a:re generalizations of penmutation- isomorphism fo¡ which canonical- fonms ar.e known, and these can be used to obtain negative results. For example, ¡re might con- side:r o::dinary equivalence of matnices. Two mabrices A and B are callçd equivalent if ther"e exist non-singular' .matrices P and Q such that B = PAQ. This rel-ation is an equivalence r€lation, and a canonical fol?n is known. Every matr.ix is 59 equival-ent to a rnatlix containing . ¡ lrs on the main diagonal, and Ots elsewhere. The quantÍty ¡ is invariant, and is call-ed

the r"a¡k of, the mainix. Thus, kre see that tv.ro nat¡ices are equivãlent if and on1y if they have the same dimensions and ¡ank. By Theorem 4.1.3, peï,nutation natr"ices a::e non-singufar, hence, if A is pernutation-isomorçrhi c to B, then A is equivalent to B. Equivalently, if A is not equivalent to B, then A is not perinutation-isomorphic to B, This does not l-ead to any useful results, since, if A is the íncidence natrix of a (v, b, r, k, À) b.Iock design, the proof of Fisherrs inequality (v < b) yields incidentalLy the fact that A has rank v. Thus, all incidence matrices for baLanced in.orpf.t" block designs with v varieties aï€ equivalent. This can not possibly he.Ip us. The technique, however", can be applieá in a¡lother" case.

fn r^rhat foJ.lor^rs, r4¡e a::e essentially eonside::ing equi_ valence of natrices whose enÈïíes 1Íe ín a conmut€.tive ring ürith cancellation. The tnaditional application of these t:esults is

to natl:ices with potynomial ent¡Íes fo" the purpose of ¿leter_ mining invar.iant factors. We shal-l state and prove our ¡esults vrith respect to the ring of integers.

Let A and.B be n x n natrices. We say that

.{ is integen-equivalent to B if. the¡e exíst two ïNS rnattices P and Q such that B = pAQ. One inmediately obtains

Lerma 4.1.8. fnteger equivalence is an equivalence reLation,

An elementa4¡ integer opet:ation on a matr:ix A is one of the foll-owing: 50

i) intêrchanging trro ¡ows of A, Íi) multip].ying a :row of A by -1,

iíi) adding an integer muLtíple of one ]1o$, to another rovr. or any similar colunrr opellation.

Lemma 4.1.9. No$r, let an elementary ::ow ope:ration E be speeified. Then there is a r:nique INS matrix P such that E(A) = PA , that is, elenentarlr tow oper"ations can be pe:r- fo::med by premultipli cation by a suitable mat¡ix. Dually, el-ementaflr colunn operations can be penforrned by a post- rnuLtiptication such as E(A) = AP . In eíthen case, the desireil matrix is given by P = E(I)

Conol-lar5¡ 4.1.10, If B is obtained fi:orn A by a sequence of elementar5¡ ope¡ations, then A and B are intege:: equivalent.

Nov¡, let A be a fixed n x n mat::ix rarith integer entries. Let ek = ek(A) be the greatest common

.O=1and ek=0 forr

Lemna 4.1.11. (i) t 0 0 < k < n . "k , (ii) er, = laet al (iii) el = scd (aIÌ a..). (iv).t-rl.t,1

He now define {=eU/e*_r,1skln and 4-=O,n

(i) Lemma 4.1.12. 9.t0,1

Proof: Let r = ::ank A. ff ¡ = 0, we are finished. otherwise, let d, be the greatest co¡r¡rnon divisor of the elements of A. Now,' let a..rl be the non-zero el-ernent r¡hi ch is smallest in absoLute value. Place this elenent in posítion (1, 1) by ¡ow and column permutations, and multiply ::ow 1by -1 if a.. < 0. Ther:e a::e nów trvo cases, according as .11 = d1 or .11 t d1. fn the fi¡:st case, by adding approp::iate multiples of::ow 1 to othe:r rows, we may r.edu ce all entríes Ín.col-unn I except all to zeroes. SÍní1ar1y, the same pro- 62

cedure nay cLear row 1 to ors, and we have A conve¡ted to

d1 t. whe::e d, divides every el-ement of Ar.

In the second case, we show that we can ::eplace afl by a smal1e:r number:. Therê ar€ tÍ,¡o subcases, acconding as a' does not divide every elenent of row 1 and colurnn 1 evenly, or does. In the fil:s t s'ubcase, l'et .ij q * o, where = "11 0 < r < ar, . Then, subtr:acting q times column 1 fuom column j pncduces a rnatrix containing r:, and we have acconplished a t€duction. In the second subcaseo we may clea:r a.l_1 of rþrí 1 and coLunn 1to Ots (except, of cour"se, arr). As r11 r d1 , and, by Leruna 4.1.13, d, is the gl:eatest common divisor of the entries in the maürix, ther€ is an entï)¡ a.. such that

.ff Átlding r:ow i to ::ow 1 Leaves all unaltered, and I "ij places the val-ue a.. into position (f, j). lle may acconplish a neduction as in the fi::st subcase. Aftel? ::epeating this process a finite numbe:r of times, ne eventually find that

d1 and the fi::st case appLies. Thus, A may "11 = , always be :¡educed to

whe::e d, divides every element of Ar.

Now, Iet d2 be the greatest common divisor: of Ar.

CIe ar.l¡r UrlU, Evidently, repeating this procedune yieJ-ds 63

E

Cor:ollar5¡ 4.1.15. Tf D = diag (dt, ..., dk) = S(A) , then tr-tlt,1

CorÐllaС 4.1.16. A is integer.-equivalent to B if and only if S(A) = S(B) , that is, the Smith Normal Fonm is a canonical-

fonm fon integer equivalence .

te¡r¡na 4.1.17. If A is pennutation isomorphic to B, then A is integen equivalent to B.

In terms of BIBDrs, this result may be stated as:

lf A and B a¡e the incidence rìatrices for isonorphic designs, then S(A) = S(¡). orl

If S(A) / S(B), then A and B r€p¡esent diffe::ent designs.

Unlike the rank, the Smith No¡ma1 Form wil]. se?ve to distinguish among designs with the same pal?ametel?s. We now pr€sent some :results conce::níng the invariant factors of the incidence Í¡atnix of a balanced incomplete block design. In the followirig, .I is a rnab:ix of appropniate dimensions, al.l of whose enùries ar"e 1. 64

Theo¡em 4.1,18. Let A be the incidence matrix of a (v, k, ),) bJ-ock design, and fet n = k * l. Let the invariant factors of

Abe {ar,...r.r}. Then êr.- 1 =D,and a.,, = nk/(n, k) Proof: Let B = kAT - l,J. Since AAT=nI+À.I

and A.t = k.I r then AB=kAAT-tA.T

= knl + k¡,J - kÀ,] = knl

Let D = diag (ar, ...0 a.r) be the Smith Norma1 Form of A, and let P and Q be the INS matrices such that D = PAQ. Then -1 -4* _'l Q -BP = knD - . Let brr ..., b, be the invariant factors of B. The matrix Q-1np-1 = kno-1 has integen entries, is diagonal, and every eLement on the diagonal divides the preceding one. Thus, knD-1 is the Smith Normal Fonm of B with its entries in reversed orden, that is

ti b.r+l-i = kn

The matnix B contains n = k - À whereven AT contains a 1, and B contains -À whereven AT contains 0. Since both situations occulS,

b, =. gcd (n, À) = gqd (k, n) , and av = kn/(k, n) tíkewise.- to obtain av_1 - , we must show O, = *. We knovr that btbZ is the gcd of all 2 x 2 subrnab:ices of B. The::e are ù 2' = !6 ÞossibLe 2 x 2 submatrices. In the cases where all 65

entries ar€ the sane Cn or -ì), the dete¡¡ninant is 0. If the::e is one n and three -)..is, the determinant is tÀk. If

there are two nrs and tv¡o -lts, the detexminant is either O aô o:: l(n'- ),') = tk(n-l). Finall-y, three nrs and one -l yield tkn. Thus 'o' : :':"i,;,i:;,':,*' k gcd (¡,, n) = kbt

Thus br=k,and i.._t = r. E

This theo:rem is very powerful, for it may greatLy rreduce the amount of wonk r.equi:red to calculate the invariant factors of the ¡ratrix at a balanced inconplete block design.

Theo¡en 4.1.19. rf n = p2 , th. invariant factors fo:: a (v, k, À) btock design ar€ completel-y dete¡mined. once the numbe¡ of 1rs in the Smith No¡mal Fo:rm is known.

Pn¡of: Let the invariant factons be a1, ..., av Ìfe knor¡ t that av = nk/(n, k) and arr_1 = n = p' From the fact that

"il"rr-t ' í3v-1 , we must have t "i=1rp¡orp-, i

Now (n,k)=pa, O

Hence k otrl a ... av_1 = p "1.2 l-_l_kp "

v+a-3

t Let 1, p, and p- occur respectively o, ß, and y times in

...r ê__ . ar, " Then a+ß+y=v-1, 9+2'y=v+a-3.

As v, a, and a are al-f kno¡m quantities, these equations have a uníque solution for $ and T, namefy ß=v+!-a-2a Y=a-2+a.

We shaLl- say that the Smith cLass of a balanced

incomplete block design is the number of 1rs among the inva:ri-

ant factors. For each of the block designs r^¡hi ch we have const?ucted, we have n = 9. Thus,

Coroll-ar1¡ 4.1.20. The invariant factors of a block design with pa:.ameter.s (36, 15, 6), (45, t2,3), or (50, 11 , 2) are com- pJ.etely deter¡nined by its Snith cJ-ass.

Theo¡em +.t.2t, If two block designs with the seme parameters J-ie in different Snith cl-asses, they are not isono4>hic.

We shall close this sectíon by stating Èhe following

theolen without prciof .

Theoren 4.t.22, Let A be an n x n mahrix of Ors and 1rs. Then

the nunber of 1rs among the inva¡iant factors of A is at least 67 L + LLoE, nJ, provided A Ís non-singular.

Ile can combine Theorem 4.L.22 and the conditÍon

ß =v+ I - a- 2a > 0 of Theorem 4.1.19 to obtain bounds on the Snith classes of our designs.

Theoren 4.1.23. LeL a be the Snith class of a (v, k, ì,) block design. If (v,k,r)=(36, rs,6), 6

54.2 - Generalized Intersection Numbers

It is r,¡ell-known that, lf A be the i.ncidence Då.tïix for a (v, k, l) block design, then AT is likewise rhe matríx of a (v, k, l) design, cal-led the dual design, The next resul-rs, 4.2.L - 4.2.4, are due ro the auÈhor.

Lemr- 4.2.1. If A is permutation isonorphic to B , Êhen

A' is pernutation isomorphic to BT .

Ite shall say rhat A is self-dual if A is permutâtion ísomorphíc to AT . We shal1 say that A is strongly self-dual if A is permutation Ísonorphic to ê s)mmetrÍc matríx.

Lema 4.2.2.. If A is strofìgly self-dual, rhen A is self_ dual .

Proof: Let S PAQ p = , where S is synmetric, and and Q 68 . åre per¡Ìrutation natrices. rhen ST = qTATfT = S Thus req qrr,rrr (Qp)A(Qp) = , or Ar = E

Lemna 4.2.3. A is strongly self-dual if and only Íf rhere is Ê perEutation naËrix p such that AT = pAp .

Ir is knor,m (c.f. Marshalt Ha1l [16]) rhar a

s)¡metric block design need not be self_dual. It is an open question ûhether a design nay be self-dual rrithout being strongly seJ-f-dual, a¡d we sha11 regard this as possible.

Theoren 4.2.4. A balanced incomplete block desígn arising fron an abelian group difference set is strongly self_duaL.

Proof: The customary construcËíon of the lncidence uatrÍx doeê not yíe1d Èhe resulÈ directly. However, if a row_ permuËaÈion is introduced r.rhích interchânges the îor,r7s coÌres_ Ponding to g and -g , Ëhe result is a syrmetric matrix, since the condirion gj - ei € D Èhen beeomes gj + gÍ é D. g

It 1s apparenr that S(A) = g laT¡ , so tesring Ínvari-

ant factora is not a final determÍnation of isomorphism. The

test described in this section can sometimes disÈinguish a design fron iÈs dual. I.Ie shal1 describe our qua.ntities first in terms of a non-s¡rnmecric desÍgn.

Let A = ("i.5 )rr*b be the incÍd.encê nêtrix of a (v, b, r, k, À) block design. I.Ie defÍne rhe inrersecËion numbefs of rhe firsr liind. (c.f. Stanton and Sprotr [36]) 69 _rb -tx - ì. tri.x , where X is a subset of the j=r. ¡!A "r.rJ

rows, and dually,

v t Í ==l ¿_,, ,t.ì.X .i5 , where X is subset of the columns. . \ r=l

Lemna 4.2.5, (i) kø=u.

(ii) tø = b

If X is a sÍngLeton, (iii) rX=r,and (iv) ç=k. If X Ís a doubleton, (v) tx=l'

These nr¡mbers clearly are intrinsic to a design, and noÈ dependent upon a paÌticular incídence natrix. They are noÈ particularly convenient to use fot classifyfng designs, as the nr¡nber of values Lnvolved, respectívely 2v and. 2b , "r" large. They are cal-led intersection n¡¡mbers because, for example, k1t,z,:] ís the nunber of elements in Bl n 82 n 83

We no¡¡¡ define intersection numbers of the second kind, which ¡,¡i11 also serve å.s invariants , and are f er.rer in number. These definiÈions and the results 4.2.6 _ 4.2.11 cone from Dulmage and Mendelsohn [14i, and were established j-ndepen_ dentLy by the aurhor. 70

Ri = Èhe number of X such that X is an n-subser of varÍeties, and rX = r . Dua1ly,

Ki = Èhe number of X such that X is an n-subser of blocks and k '= rn .

1 Thus, R; counts the number of unordered trlples of varietíes which occur together exactly twÍce.

Theoren 4..2 .6 . (i) R0=1if m m=b, =0 if nlb. r0=l if m [=v, =0 Íf nlv.

(ii) Rl=v Íf [= t m t =0 iÍ n+r rl=b if m n=k, =0 if n*k

(iii) *:= (;) m= À, " =0 if nt À

Proof: (i) There is one o-subsêt, nanely, the enpty set.

Our values fol-low from tø = b, kø = u .

(ii) There are v l-sets of varieties, and each con-

Èains r e.I-ements. I{ence, we obtain the values ot .mmn] . K] is obraíned i.n dual fashion.

(iii) ln similar fashion, rhere are (ï) ,-""." .t varieties, and each 2-set occurs in I blocks. g 7L theore¡n 4.2.7. (i) .For n> k, ( pD='mt I (l) i"=o 0 if nl0

Torn>r, I t ,."- ) (l) * '=' r- 0 if nl0 I

(ii¡ For n > .1 and Ð>r,*i=0. I'or n> I and n>k,*Ï=0.

Proof: (i) If t1 2 k , Lêt X be an n-set of varieties. ClearLy .X = 0 , sínce every terü in Èhe sum

for rX contaíns a factor of 0 . The duaL resuLt holds for blocks. (ii) These conditions reflect the fact that ê ser of blocks can share at most the k varieÈies in any one of then, and the d.ua1 condÍtion for

varieties .

Theoren 4.2.8. Tf n>2 and m>À,then *T=0. ¡or a s)rmetríc design, Lhe same resulÈs hold for Ka .

t.a N"a be the nunber of s x t ninors consisting of aLl ones in the incidence oatri¡r A . Fo¡ s=0, weshall take Èhis to nean andforÈ=0 72 Theore¡n 4.2.9.

= , /'\ *;." "". J" (;/ È¡ 1'l *' m>s \s/ m Proof: Fix the integer m . There are nfl Ia7.f" to choose s rorrs so Èhat Lhere are exactLy m col-umns consisting entireLy of lrs. ¡'ron the m eolunns, ïre nay notr make a selection Ín

,/ m\ (t) *"y". Summed over all vaLues of m , rüe obtain

n-'st ='- ,n /n\ onos ( t/ A. sinilar argunent applied to'columns yÍel_ds the du¿l result. El

Corollary 4.2.10. (i) (ii) "-i=(;) ', -i = (l) (iii) ,, *l = , (i) , (iv) , *"='/t\ n n \n/ (v) r¡o ü) -i = (;) (l)

Proof : (Ð. is the asserrion = \, (;) : i¡\ Duall-y, (ii) is Ëhe resulr rhâr NOn = (;/

.(ili) We require Nrr, Since there are

b ways of pÍcking the coluûn, and ¡kt (;/ ways of picking rhê n roÌ¡s, ¡ùe obÈain "",=o(:) . 73 (iv) is dual ro (iii). úe.r""

. ","="(ï) (v) I,Ie require Nrr, . There are (î)

ways of seLecting Èrío roT{s.

This gives À colurnns fron which

n must be selected. Thus

-," = (;Xl) .

CorolJ.ary 4.2.LI. Ior a s)¡nmetric design,

(î) .i = (;) (l) " The uÈility of considerÍng inÈersection numbers of the secon

Theorern 4.2.12. The intersection numbers of the second kind alrd the numbers Nst are invariant under row and column permutations of the incidence natïíx A .

Equivalently, Ìüe obtêin

Theoren 4.2.13. If two designs dÍffer in any intersection nunber of the second kind, they are not isonorphic.

lle sha1l eurploy the results of this chapter to give computer tests for isomorphism of designs. our results are discussed in Chapter five. 74

Chapter 5 - Algorithrnie Techniques in Block DesÍgns

In this chapter r¿e discuss a varieÈy of topics. First, r,re describe computer programs for constructÍng incidence matrices for the three designs developed ín Chapters Tr¿o and

Three. Second, r¿¡e survey the Literature for other occurrences of designs with the parameters (36, f5, 6), (45, 12, 3), and

(56, 11, 2), and provÍde a tabularion of Èhese desÍgns. Third, we apply the theory of invariant factors and of generalized Íntersection numbers as described in ChapÈer Four to deterrnine nearly completely the isomorphisns whÍch exist among the knorrn designs.

55.1 - Computer Construction of Incidence MatrÍces 55.1.1 - The (56, 11, 2) Design

The plane PG(z, 4) is an example of a (21, 5, l) block design. It is rrell-knohln that the difference ser

K = {0, 7, 4, 14, 16 oodulo 2t} describes Pc(2, 4), and Ít Ís Ëhis descrÍptíon which we use in our conputer program; Pc(z, 4) is specified by a set . 75

{P0, Pl, ..., P ZO! of pdÍnts, a ser {LO, Ll, ..., L2O} of lines, and the incidence rule

P__xy is on L_- if and only if x * y e Í( ,

Ëhe additÍon being perforrned nodulo 21.

Lemna 5.1.1. There are two unique functions f and g rriÈh

range {1, ,.., 20} and domain K such Èhar

n=f(n)-g(n) (nodulo 21).

Proof: This is jusÈ another way of sayiag that K forns a (21, 5, l) difference set.

In aûy computer progrãu for pG(2, 4) we shal-l need

the function h(x,y) which produces the subscrlpt z of the

1Íne L" r,rhÍch passes through P* and p" . Ì,¡e noLe that, because of the form in r.¡hich we have stated the incid.ence ru1e, the funcÈion h also eomputes the subscript for the poÍnt coDmon to a pair of lines.

Leuma 5.1.2. h(x,y) = f(x - y) - x (noduLo 2l) = g(x - y) - y (noduto 2t).

Proof: Lex Lz be the line through p* and p" . Then r¿e have x+z=dreK,and !*z=dreK Subtracting, rùe obtain x - y = dt - dZ As d, and dZ are both in K , they musÈ be the unique components of the dÍfference x-y. Then dr=f(x-y) and dr=S(x-1,) Our resulrs êre iüûediate. 76 To illustrate, I,re shall- compute the three ovals

Lhrough three flxed points (cf. Table 2.2.1). We begin by

forming a table of the differences of pairs in K . We then use these differences to tabulate the functions f and g

T¿ble 5"1.1 - Differences (x - y) fron K

I7 18

10

T2

TabJ.e 5.1.2 - f(x) and e(x)

Ite 6rart by selecrÍng tO,, Þ1, and p, . We have h(r,O) = 0, hC2,0) = 14, an¿ rr(â;f) - zO . rhue, we see that the polnts are not collinea¡, but rather forx0 a triangle ¡¡hose sides are LO, I,rO, and. LrO . TabulatÍng thè points on theae three lines, rúe get,l .77

L0 = {Po, Pl, P4, Pr4, Pro} ,

. {P0, P2, P7, Pg, Ptl} and "t4 = Lzo = lPI, P2, ?5, P15, Pt7] .

None of Ëhe tïre1ve points in thÍs list may be used to augnent

our trían;le to a quadrangle, P, is an al-1o¡¡able point. Computing again, vre obtain h(3,0) = 1, h(3,1) = f3, h(3r2) = 19 Thus, ve have a compJ-ete quadrangle P0, Pl, P2, P3 , whose sides, written i"n opposite pairs, are L0 Llgi I13 "od and LrOi and L, and LrO . Usiog h again, we get h(19,0) = 16, h(14,r3) = 8, and h(20,1) = t5

The diagonal points for the quadrangle êre thus pg, ?tS, and

PtO As we have h(15,8) = h(16,8) = h(f6,15) = 6 , we see that L6 are Pr P16, Prr, P1U, and Pl9 . Excluding rhe three diagonal poinÈs, rre find rhat {p0, pl, pZ, p!, ptO, tt9} ls an ova1. Sirnilarly, selecÈing p6 to compl_ete a quadrangle gives ríse to the oval {P0, ?f, pZ, p6, plZ, tfS} . Final1y, choosing P, yields {?0, Pl, ,2, P9, p2O} . Denore "tg, these three ovals respectíve1y by A, B, and C .

. The program for computing the incídence rnatrix begins by cornputing all quadrangles. Tor each quadrangle, the diagonal points, the dÍagonal line, and fina1Iy, Ëhe tt/o non-diagonal . points on the diagonal line are computed. These tr,¡o points complete the quadrangJ-e to an oval, As each oval is obtained, it is conpared against the three known ovals A, B, aad C

The conputed oval is placed ï.nto one of three.lists, depending on ¡,¡hich one of À, B, or C it meets an even number of times. 78 After aL1 of the quadrangles have been computed, each of the lists contains 56 ovals. The program then computes the incidence natrix for the (56, i.1, 2) design directLy froru the defínirion gi.ven in â2.3. 0n1y those ovals which neet A an even nut[ber of tines are used. The rovs and col-umns of Ëhe incidence mâ.trix eorrespond to these ovals in the order in r¡hÍch they arise . during the first stage of the pîogram. This happens to be

lhe lexicographic order which arises from our numbering schene for the poinrs of pc(2,4),

Fina11y, the incÍdence natrix, wi th an apptopríate

heading, is szritten onËo an externaL storage nedium. The matrLx ¡,¡i11 be examíned later to verÍfy that a BIBD has been produced, and to calculaÈe sone of the lnvariants associ.ated

$/ith Èhe design (cf. 55.3). The progran Ls lisLed on pâges A.1-A.3'andthedesignonpages3¡12-B.l3.'.'

55.1.2 - The Ðouble-Six Design

lfhen dealing with the cubíc surface designs, we r¡i11 use the inÈegers I, Z, ...,27 to denoÈe the line.i:

Lemna 5.1.3. The function c(i,i) = i x (1r - r)/2 + J + 6 nap6 .{(i,i) I f = i. J < 6} biuniquety onro {13, ..., Z7}.

lle then å.dopt the Ìepresentation a. + L i b. -t 6+i, and J 79 ci¡ * .(i'5¡, r,rhere i < j

The startifrg poiat for the progratr Ís Theorem 3.3.5.

I{e prepare a lÍst of the 36 double-sÍxes as enumerâted. there. First, Èhe original double-six

l^, "r. ", ^4 ^s ^6 o, o, b4 bs bo l.o, is recorded. Next, the 20 double-sixes characterized by rì tse l"r ^2 "3 "40 "¿sl ll b+ o, ou l. "z: "rr "rz J

âre recorded, and fina1ly, the 15 doubl-e-si*e" .h"r."terÍzed by tìtt bl t23 I "24 "zs "zo I tt bz |.", "r3 "r4 "rs "ra j are reeorded.

After the l-ist of doubl-e-sixes is conplete, Èhe incidence matrix for the (36, 15, 6) design is computed.. The defínÍtion of = given ín 93.3 is used. A.s in the case of the (56, 11, 2) desígn, the Íncidence matrix is then sËored for later use. The program is listed on pages A.4 - A.6, and the design on page 8.5.

55.1.3 The (45, L2, 3) Design

In this program, r,re begin by constructing the incidence natrÍx for the lines of the conplete double-six, accordiag to Theorem 3.3.1. lle then consider all ,"r', i

'80

unordered paÍrs of lin.s. Wh.o.ver the 1Ínes neet, r¡e find. the third line !¡hi-ch neets both, and enter the three into a table

ot the triËangent planes. I'inally, I,lre compute and save the

ineidence ¡natrix ior !h" 14S, 12, 3) design, using rhe

definition of = fron 53.4. The program is listed on pages À.7-A.8, and the design on page 8.10.

55.2 - Other Constïuctions for the Designs

In Èhis section, rre tabulate additional descripÈions

of the deslgns with parameters (36, f5, 6), (45, 12,3), and

(56, fI, 2). If the presenrarion is made by a difference ser, the difference set is included here. Otheñ,¡ise, only a bríef conmenL as to the nature of the derivatÍon is included.

Programs to conpute i.ncidence matrices for all of the dêsígns Listed here nay be found in an appendix. The presentation of this secti.on is chronological-.

Ta 1962, K. Takeuchi 140, 4fl gave di.fference sets for both (36, f5, 6) and (45, f2, 3). ite shal1 refer ro rhese desÍgns as KT36 and KT45 . The difference sets are

{000,010,020, 100, 110, 120,001, 101,.

. 2Ol, OO2, lI2, 222, OO3, I23, Zl3 nodulo 334], and .{oor, ott, o2L, ooz, Lo2, 2oz, 003, 113, 223, 004, 724, .Zl4 nodulo 335].

A progran túas !ürítten to produce an incidence natrix from a difference set in any Abelian group r¿hich is the dlrect sum of three cyclic groups. This program appears on page A.9, and the designs are listed on pages 8.3 and 8.8. .81

In 7962, Shrikhande and Singh l34l shor¡ that BIBD's

can be derived fron fi¡o-class association schemes. I,le describe their method in detail, as r¿e sha1l return Èo iÈ ultimately in settlir¡g rhe (45, 12, 3) isonorphisms. Alrhough associatÍon

schemes nay be defined. I,rith a larger nunber of classes, we shall- only discuss the ÈÍro-class case here.

A two-class assocÍation schen¡e on v varieties is gj-ven when (i) "n, ,,r.rffi"" r= said to be either first or second associates; and (fi) each variety-I has n, first associates and n. second ¿ associates; and (iii) if an (ordered) pair of disri.nct varieties are i.-th associates, . then there are p1, varieties which - llc jth assocíates of the first and kth åssociates of the second ariety, .for l

Theoren 5.2.1. (Shríkhande and Singh [34]) (a; La . P3¡ = Pt¡ r

Theoren 5.2.2. If there is a two-cLass association schenè with parameters v, nl¡tz n¡r and pir-, 1< i, j, k < 2 such that 1 t JN Pif = pil , then there is a s rrongty-self -d.uat BIBD wirh 82

Parameters v, k = nl, t - ni, =.n1, . Proof: Define blocks Bi, 1 .. 'i I v so that block B. contains all first associates,., 'of Vi Each block contains n. varieties. The intersec{ion of distinct blocks B. and oj-'r2'.a contains pii or pr'a varietÍes, according as V_ and L Uj aïe first or second associates. As these m¡mbers are equal, Èwo distincr blocks inrersecr , in exacrly I = pif = pîi elements. Consequently, our incidence structure is the dual of a symetric BIBD rüi th parameters v, k = nr, À, and. heace a BIBD of those parameters. Thât the BIBD i6 strongly self_ dual- foLlows fron'Èhe fact thãt O, . Uj if and only if V. and V, are first associates if and only if V. e B. g

Corollary 5.2.3, If a two-class associaÈion scheme has pZZ72 piZ = , then rhere Ís a s trongJ.y-se1f_dua1 BIBD r,7ith parareters v, k = nr, = nLz= . ^ nî, , In descrÍbing a Ël¡o_cLass âssocÍation schene, it is cusËomary to !Írite the parameter" p-it a" n¿rtrices liplr pizí\ 't-[n*p. =l i ìI ' =t'2' \ 11 "r/ In 1954, Bose, ClaÈnorLh),, and Shrikhande l9J presented a two-class asaociaÈion schene lrith v = 45, n, = 12, îZ _ 32, 13 8\ t3 e\ and Pr = *r= (, ,r/ ' \, ,r)' Theoren 5.2.2 appj.ies to yircld a (45, L2,3) desÍgn, whÍch we dênoÈe by SS45.

A piog,ran was writËen to corivert certaiû trüo_c1ass 83

association schemes into: (45, 12, 3) designs. This prograrnrne appears oÌr page 4.10, and the associated design on page B.ll.

Shrikhande and Singh refer Lo the folJ-owing theorem of Bose and Shinanoto l8l:

Theoren 5.2.4: If there exists a set of (t - 2) routualJ_y orÈhogonaL Latin squares of side 2t, Èhen there exists a two- t class associaEion scheme with v = 4t', îL = t(2t - 1), and l2 Plr=Plt=t(t-r).

0f course, the other parameters are al_so mentioned, but we oûri t then here as they are not necessary for our purposes. They nay be easily deduced from theoren 5.2.2. An imediate corollary of theorems 5.2.2 a¡d 5.2.4 ís..

Theoren 5.2.5: If there exists a set of (t - 2) nutual-ly

orÈhogonal LaÈin squares of side 2t, then there exists a balance lncomplete block design r{ith paramete rc u = 4t2, k = t(2r - 1), and À = t(t - t).

Shrikhande and Singh state Èhis result, which has been obtaj-ned Índependently by the author. It is imed.íate that Èhe condiËions of the theoren hold for t = 3, and this yields a (36, f5, 6) desÍgn. lte shal1 return ro rhis design when we discuss the work of Blackwelder belor¡. In 1963, a (36, 15, 6) design was prinred in tl6l. The design, attÌibuted Èo p. K. Menon, rde ghal1 ca1L plf36. It ls given as

{11, 12, 74, 76, 2I, 24, 26, 35, 41, 42, 46, 53, 6I, 62, 64 nodulo 66Ì. prograrû A for converting (6,6) diffe¡ence sets into (36, f5, 6) 84 . designs liTas lrTritten. This piogran appears on page A.11, and the design on page 8.4.

In 1966, Bose and ChakravarÈi [10] constructed. a tr¡o-class associatÍon schene fron the points on a cubi.c surface in PG(3,4). Using the nethod of rheoren 5.2.2 trley deríve a (45, 12, 3) design whÍeh r¡e call R845. Using the sane prograû as for ss45, r,Ie obtain the alesign listed on page 8.9.

In 1969, J. !,IaLlis [40J presented borh a (36, 15, 6) and a (45, 12, 3) by ltsrlng the incidence naËrices. The larter design 1s given incorrêctly, and should be corrected as ior-lows: change the 9 x 9 ninor _ in rows 37 45 and colunns 27 _ 36

from (: "i) (: iT) : This should be regard.ed as a correctÍon to a transcription error in the source text. trrle shalL refer to these designs as Jhl36 and JtrI45 respectiveLy. prograÌs for converting Ëhe abbreviated descripËÍon6 in Èhe source to íncidence nêtrices are given on pages A. 12 _ A.15, and the designs are listed on pages 8.2 and 8.7.

In 1969, !f. Blaclcwelder t5,6Lpresented. the construcËion theoren of 5.2.2. From the two-class êssoclatíon schene of 3ose, CLaÈworÈhy and Shrikhande he deriveLatin square, tàcftLy using Èheorems 5.2,2, 5.2.4, and 5.2.5. tte sháL1 caLl rhis desÍgn tr{836. 85

The author has shown an", an" design ltB36 uray be real-ized as

a difference set. To see this, r"¡e first explaiu 3lackr¡elder I s construction. Let A (a_r) be the 6 x 6l,atin squarà = r-J defined by j (mod arjrJ =í+ 6), for0

Tsro distincË ordered pairs (Í,j) and (k,L)'are fj.rst assocÍaLes if i) i=k,or 1i) j=1,or

1ií) 4..r-J = €Lr' Theorem 5.2.6: The design WB36 Ís isonorphic Èo the design

given by the difference ser {01,02,03,04,05, 10,20, 30,40,50, L5,24,33, 42,55 noduLo 66]. Proof: !üe have an iruoedÍate ídentification of the varieties r¡ith our group elements. We nor,7 interpret Ëhe three ori-ginal conditions on varieties as group condítions on the dífference d = (i,j) - (k,1) = (i - k, j - 1). The firsr and second. condÍÈions are equival-ent respecti.vely to d = (0,x) and ¿ = (x,O), where, i-n either case, x I 0. If the third case applj-es,

¡¿e have a.. = a- r¡hence r-J t

PG(3,4), and MII56 from ovals in pc(2,4). In borh cases, the development is viê permutation groups, not geometry.f In 1971, E. Spence f35l shor,¡ed rhar a (36, 15, 6) design nay be derived from a Hadarnard maÈrix of side 36. A ptogram for performing Èhis constructíon Ís listed on page A. 16, and the design, ¡vhich r,re cal_1 ES36, is on page B.l.

I'ina11y, we shall use the names RK36, RK45, and RK56 to desÍgnaÈe the designs derived. Ín Chapters Two and Three of thÍs thesis. l^Ie see Ëhêt we have six designs nith paraneters (36, 15, 6): ES36, JI^I36, KT36, pr,f36, RK36, and !ü836; we have seven designs $ri. th parameÈers (45, L2,3)z Jl.I45, KT45, MIl45, R845, RK45, SS45, and WB45; and rre have two designs with parameters

(56, I1, 2): Mlt56 and RK56.

In a private cornmunication, A. Rudvalis states that he has obtained a (56, 11, 2) and a (36, 15, 6), and rhar Hall- h¿s obtained a (36, 15, 6). These designs iráve aI1 been obtai¡ed by group- theore ti"c methods. nudvalis (56, states thaÈ hÍs 11, 2) is isomorphic i,rTirh rhar of ita11 (tßt56) , but thaÈ the (36, 15, 6) desÍgns are not isomorphic. We l¿il1 not deal with these designs in this thesis. 87 55.3 - fsonorphisrns among the Designs

Fo¡ the designs with paranéter"s (56, !!, 2), we have noted that both are der:íved from the ovals of pc(2r4),

although no details of the derivafion for" MH56 are given.

However:, Jonsson P2.l gives the geometric d.erivation of MH56, and this work shows that the designs are the same. Thus, in

the case of parameters (56, 11, 2), we have one design.

,In the (45, 12, 3) case, there ane tr.Io obvious

isono::phisms among the seven designs, namely MH4S vrith RB4S,

and SS45 with W845. The possible number" of distinct designs

is the::eby ¡educed to five. We shall see that thene are at l-east th:ree non-isomorphie designs ar¡ong the five.

Among the six (36, 15, 6) designs, there al?e no

obvious isono:-phisns. We shall see that there a?e at least

four non-isornonphic designs among the six.

We eÍìp1oy a conpute¡ prog?an to calcul,ate some of the invariants described in Chapter: Four. First, the inter:section I nurìbens R" (0 g m < tr) are computed. (For the ::emainde¡ of this

section, we shal-l dt"op the supe¡script, and write Rm, ) Duning the computation of these quantities, it is convenient to test

that the now (variety) restrictions for" a BIBD are satisfied, and this is done. After these values have been calculated,

the íncidence matrix is t:ransposed, and the process ?epêated.

ff the incidence mat:rix is symnetric, or if the design is known

to be strongly self-dual. (as, fo:: example, is the case when the design arises irom a difference set), the latter step is 88 sup?esse;. Fina1ly, the Smith class is computed.'for eaeh design. In fact, this last calculation ís not a fuLl computation of the invariant factors. We continue reduction of the incidence mat?ix, and consequent counting of 1rs among the invariant factors, for as long as the natrix contains an entry of t1. When all ,of these have been exhausted, the ::emaining entries are.tested for divisibility by three, as this is the indication that we have completely d.ete:rmined the invariant factol.s. It is entirely possible that r¡e níght not reach thís stage. Hora'eve?, in practice, it tu¡rns out that ES36 is the on.ly design which does not run to completíon, and other factots make it unnecessary to pu?sue the calculation in this case. The progran is listed on pages 4.17 - A,21 . The mat¡ices l.lB45, MH45, anC MH56 wel?e not tested, as they are known to be isomo¡¡hic to othelrs among the designs, In eve::y case, the desígn and its dual turned out to have the same row (va?iety) structut:e. We might note that the numbe¡s

Rm fon the (56, 11, 2) design can be calculated $¡ithout computation. l{e demonst::ate this, and pr"ovide check values for the other cases in

Theorem 5.3.1. (i) For. the (36, 15, 6) design, r^re have Eþ-R 7r4o- m=0 m =

_6 r*=1 ffi* 16380 and = '

t"-2-6 (r)Rm.n = r27oo. 89

. (ii) Fo¡ the (45, fZ, 3) designs, we have t3 R rì^n 'm=0 "rn = "

om=1i3 nR_ m = 9900:' and

E"-2 (ä)Rm : eso.

(iii) For the (56, 11 , 2) designs, R^ 181+80, U = "7R = or4O, and R- 0. 2 = Pr.oof: Éut n = 3 in (i) and (ii) of Theorem 4.2.!o, and. also in Coroliar5r 4.2.ff , fn the first two cases, this gives the stated results. In the case of (56, 7!,2), lre get R2 = 0 frorn Cor:o1l-a¡v 4.2.!I, and the other two conditions enabl-e us to sol-ve for RO and R1. We could also have had R2 = O as a consequence of Theorem 2.2.!g. g

Table 5.3. 1 - fnvariants of the (36, 15, 6) Designs SMTTH DESTGN Ro *tl*, ll l*rloul*rlR6 I ctASS

ES36 0 846 2742 0 >16

JW36 144 468 3996 2232 288 0 t2 72

KT 36 648 3888 2196 324 0 I4

PM36 144 468 3996 288 0 T2 t2

RK36 0 540 4320 2160 0 0 L20 15 I,lB3 6 l.-; 468 3996 288 0 72 90

Table 5.3.2 - fnvar"iants of the (45, 12, 3) Designs

SMITH DESIGN R^ S

,JW45 5220 810 0 810 60 18

KT45 5220 810 0 810 60 t7

RB45 5040 8640 270 240 _t3

RK45 5040 8640 270 240 _t :)

SS45 5040 8640 270 2+0 15

he design RK56 has a STnith class of 20,

We can novr dr"aw some conclusions, based on Tables

5.3.1 and 5,3.2. First, the (36, 15,6) designs ES36, KT36,

and RK36 are all dístinct. Second, the designs ,IWg6, pM36, and l'¡ts3ô ane distinct from the first group of three designs,

but, of this group, any o:. all of the designs may be isomorphic.

Thus, ther.e are at teast four, and possibly as many as six, non-isomorphic designs Ì¡ith paramete¡s (36, 15, 6).

' We a]so see that the designs J!{45 and KT45 are

distinct. Among the designs RB4S, RK45, and SS45, which are

all distinct from the first two designs, we may again have some ot? all isomorphic.

lhe possible Ísomorphisns among JI,I36, PM36, and

IfB36 remain unresolved at Èhe current time, although the fact that the latter two arise from (6,6) difference sets suggests that there may be an isomorphism here at l-east. The (36, 15, 6) desigts of Ha1l and Rudvalis remaín outsÍde our classífication as weL1. 97

55.4 - Isonorphisn of RK45, R845, SS45

I,Ie now compJ-ete the discussÍon of isorrorphÍsm among

Èhe designs RK45, R845, and SS45. We shall see, in. f".t, tt"t the designs are all isonorphic, and obtain explicit per¡ûuÈa- tíons to demonstrate this.

As noted earliêr, the desígns R8.45 and SS45 are both obtained from tr¿o-class association schemes. The assocÍation schemes are presented in identieal- fashion: 27 groups of 5 varieties each are given, the varieties in each group being mutually fírst associates; each variety oócurs equally often

(3 Lines), and a pair of varieties occurs logether at most once. The program cited for creating the designs RB45 and SS45

ís designed io process association schemes of exactly ti¡is forn.

trle note nor,r that our design RK45 can also be des- cribed in this format: for each line on the cubic surface rùe note :rs a group Èhe five tríÈangent planes which contaÍn that 1ine. The association schemes for the three designs are listed in Table 5.4.1. The lÍnes of RK45 occur in the order inplied by the representation noted Ín S5.I.Z, imediately following le¡¡¡r,a 5.1.3. The numbers assigned to the

Èrítangent planes are the rot¡ and col"umn numbers ËhaË apply

Èo the incidence natrix for RK45. These numbering schemes are dÍspl-ayed explicitLy in tables 5.4.2 and 5.4.3,

I'Ie now utLlize Èab1e 5.4.1 to construct explicit l-sornorphisms beÈween RK45 and R345, and betlreen RK45 and SS45. 92

!3 ::4 :ì I 14 t9 :lC .,^ j.i E rq ^^ 1 û :-ì 1 :l:ì 33 ..!! 't f: . L 3 :! iì i)'t 2e :rg 1 2': !;0 .i': t:2 i 2e 43 rl+ 45 7 !2 ':17 r;0 r- 3 ¿ 1.7 :jq 4:. rrji I :: 35 3ú ri! 10 2'i 'J6 39 4:: 13 !'¿ 37 tiz r'l ¡ 1tl 23 32 39 4i. 15 28 33 38 rrl- i9 21 33 3ô r+3 20 29 32 35 40 2Z 3C 31 34 3'i

i'A:.] i.,; 5.4 ,': ne that the isonorphÍ-sms exist, and that the problen Ls to identify then.

We also presume thaÈ the isonorphisns rnay be recogn Ized by

treaËing the associaÈi.on schemes for RB45 and SS45 as f-ine/plane

fncidence schemes analogous to the cubie surface scheme.

Ilenceforth ¡.¡e sha11 use the terns line anil plane r,,7ith respect to the latter tr,ro desÍgns i.n the r¡ay suggested by our presumptions. Of course, we say thaÈ tr,7o lines of a scheme are incident if Lhey are incident r,rr:ith a comlon plane.

lüe deal fÍtst hTith the question of isonorphÍsm betr^/een RK45 and R845. I.Ie attenpt to find a double-sÍx among the lines of R845. Theorem 3.3.2 gives us the key to our search: find a set of five skew línes r¿ith a coumon transversal, and a double-six should appear. I,Ie begin by selecting the first line of R845, and identifying íÈ as line in rhe "l Schlllfli notation. lle iu¡medíately see that the ten lines 2 and 3, and, 4 7, l0 and 19, 14 and 15, and 26 and. 27 all neer line al ; the Lines have been written Ín incident pairs, the incidences arising out of planes 17, !, L6, 21, and 25 respectively.

Accordingly, L7e assign names b' b3, b4, b, and bU respecÈÍvely Ëo;l1n.es 2.,..4, 70,.14,. aod 26.. This has imediare impl-icarions; lines 3, 7, 19, 15, and 27 musr be respecÈiveJ_y lines ClZ, gtS,

Cl4, Cl5, and Cr6i planes 17, 7, L6, 2I, 25 in RB45 nusÈ be planes l, 2, 3, 4, 5 respectively in RK45. We bhould no¡¿ be abLe to complete our naming of the lines in RB45 accord.ing to the Schläfli schene, follor,ring the ouÈline of theorem 3.3.2. .94 Every 4-subset of ¿he lines {b2, b3, b4, b5, b6} oust have two fransversals, and another, the latter being a. where "1 , b- has been excLuded fron the 4-subseÈ, lforeover, there L musÈ be a transversal to al-l- of {ar, a, a4, a5, a.6i} and it must be sker¡ to b., i = 2, ...r 6 . Thís line is, of course, b^t . Ilaving esLãblisheal these names, r,re can proceed to name alL other lines according Èo theorem 3.3.1. Of course, lle get imediate identification of our planes.

Proceeding along these J-ines, we find thât l-ine 13 is a second transversal of {b2, b3, b4, b5} , so we name thÍs line a6 . UnfortunateJ-y, I,¡e nor¡r have problens: í) u6 and bU neet (in plane 33),

ii) none of the other 4-subsets has a second .. transversal, and

íii) there is no line sker¿ to b2, b3, b4, bS and bU

Al-1 of these probJ-ens are consistent r^riÈh a reversal of the roles of the trüo lines bU and r'7e try again, Èhis "16 i tine identifying line 27 as bU , (and Line 26 as .tO). This does not force a change in any of the other names assigned. Proceetü-ng as before, we find that lines 18, 5, 11, and 21 are respectively a5t a4t a3t and a, . Tinally, we discover that line 12 is b, , aud vre do in fact have a double-six. ConÈinulng to name lines according to theorem 3.3.2, we are led eventuall-y to a conplete double-six in the tkTenty-seven lines; as no further problems arise, r,re have 95 Theorem 5.4.1. Ihe designs tiX¿S and RB45 aie isomorphic.

A sinil,ar argument shohrs that Èhe lines / tlI t zo L3 L4 L6 t8\ \2r\ 2 4 6- 8- --/l}l form a double-six in SS45, and agai.n the nanring of the 15

auxilíary lÍnes can be compleÈed. wíthout problen. Thus

- Theorern 5.4.2. The designs RK45 and SS+S are isonorphic.

The actual isomorphi-sms are sr¡nnarized in tables 5.4.2 and. 5.4.3.

We may now conclude that Lhere are three distinct

(45, 12, 3) designs, characrerized by JI,I45, KT45, and RK45, s-c-l!4!t! I!{!2 PB t' " ,g$ri 5

A. i. 7 A2 {3 3 A4 4 Á5 5 A6 6 7 iJ2 g ii¿ 9 iì!!. 1i) i3, t7 ¡J r,' !2

c72 LJ :1 14 7 1'1 h 15 19 U TJ 1() '_a :', {/o 7'Ì :,6 c 2':. !8 :: 0 C 2:, :L: c25 2a. ..' 3 ii 2.4 ü3 5 23 JI 2i. ,?. ! :l3 .: i 2i) 1: . LJU 27 c)

:' ,1ì ), ¡.: 5 a,-a-t7.: /t- tT I 97

5'.' :!i.,|L!1., i )i': ; r) J

1 a ,{1 3 A7

:;

10 '!i L ',: 1:i i¿ itl 5 29 '! 'j 1./ ;.1 \)_ It :: c Lr)

22

:lô 27

29 j l.l c 31 7 _?2 4:l

34 45 35 36 36 4L 37 19 't i) 3

40 3l 41 3ll 42 !7 43 6 ::.4 1i r!4

0r -'ìt . :)..,,,'li,l¿," APPENDIX A

COMPUTER PROGRAMS \l!l'p/ ¡Jf ß '"¡.r,T': I ü lìfìG"l r A P. il r Àl0r XT I lr'tT-:r;;:r F',lr\tt:I i¡i ïFr?iJ (f*,rN I 1. C T I-i': í I': I}\I T!I').i' P?CDUCTS .THT i\IJMBER OF THE LJl.If TH?IlJGH c 1-'r_- Ì.1.1' pJi't'¡s :.{Hflsf t'\Ji¡Êç'ls ¡RE GIV:¡'l ARGUT4çNTS. IT I S C x 9:LF-tl..j il- FlJijiTI:i'1. {T IS P^SSD O¡l Tli-Ê^3 DIFFiRf\1C1. SÍT a, \l!r\rl.ji,! F-ì), DGI?rr. ) c3T¡, I,...lED FRil¡4 THS S¡T (0r1r4, LL¡151. C ,.2 IilT:ç':I I':)T(2J)/Ir 1:1t 14tCr LrJr Ir 4¡L4¡4tl6r2*14r2*t 5tt)¡|tl4¡O/ J THìJ=.'r.ô(lir(''j:ll(1,1-:¡.1|2lr2ll l-'!+21r2Il 4 'ì;i lt'¡ 5 ':11'l I i T t c 1o:ì \Yì..'lf-ì I I I ¡ L I Z .\ ; I I I

.tPLf ,i;., a1, r l'l: i ( \-/ ) 7 l) "li :í;:- ì: ¡(Él,Kv.<(3ì /3*t / ¡u.!Y1,55 ¡¿: tll 9 t.,lV \ L r:ìjL - ( D,rl ), l?l2l,DZ l I ( P{ 3 }IP31I (p ( r}} ,P4} I f P( 5) I P5) I \()(;)rr)r.) 11 I'l f ,,C --tì '11 Xl,'i( 5,iró) L1 :0JI'/¡t-i-',|at -:xli),H:xl .., 12. t,tlr':a 1?t..2 1ì. ì I X ( 5 j , 5 ô I / 317 5\.O / a c.:T TH: trlx.r,)s 1F ?Gl2,4t a .r"-riI p3, p4. i .¡ i ï!l- .ì,t.\f.ìÀ.,jCL r ;i ITÈt Vl:aTlC:S lLt ÐZ¡ C St ¡,lLT: ,':,LSLy l.ìIÂl l ITS SID:S: Ll=p1.D2r STC". r_ v;al--y ri;r iFt sISl:s lRf ,rt L DISTIIICT. c 13 1.i lL 1l=1r21 lrr )ì.-l l-I lj ìl LZ i:.-lir2i l'r Ir'( fZ .:'),I I ) Gf Tr:) 12 Il t>r_-i"_ I :ì 1:ì l- I = lr I u ( Þ 2 ].l I ) lÕ Il tl I1=1 2¡2_l' 2C rc(Il.:e.ì2)Gl T'l 13 ?l el=l l-t l- 2 I' l ?'J ( f' 3 Þ ì ) ??. = ' ?) iF (LZ.:G.LllGC TO 13 ?+ L3=rïtìLj(P3,P2) 25 IF ( L 3 . i C . L I . :ì :ì . L 3 . i Q . L 2I G': r3 ll 2t¿ qî 1r i 4=\ j,2L 21 Ir= (14.!'!.ll) Gl To 14 2è P:-= i 1_ I 29 L/.=rilr'u(P1-,pl) 3C I f: (t_¿..j rì. L 1.1i.. L1'. i1.L2.îD.14. !C. L3lGl-l r'l 14 3l Ll=lr-llìrj{P¿:rÞ2} 'l ?2 IF ( L r . :, G . 1- . I F . L 5 . : !. L 2 . I e . t 5.:Q.13.0Þ'.L5.8Q.14' G'l TC l4 /.= j3 L i-:.t ìiJ 1o,* , P -3 ) .-r 7/. I tr I L':, . :ri. r, t.. li .Ló. !1.1..2.np .15.:6.13.lR.L5.'Ë0.L4.OR.Ló.10. L5l X/l iÏll'+

À.1 C ' C GïT TI-" i)T¡G]JIÀL PiI.ITS .{I.I] ThE LIIIE THC,U TH!14. c. 15 ll=ïilÞu(11'Lú) 35 l2=T*lìU{12'15) 77 ì-?=rlt1,,(L?,,1,, 1 31 lL=rrl'ìU(DI 'tl2 ) a, C F ir!l Trl i¡- l,:i l-DI;,Çf rl¡rL Plll.iTS C\ THE )I AG1NAL L INç. c Tl.::s: Þ.r l1':l s c:l'1eL:I:: Tlr: qrxA^. THES: POINIS ARE FrìUfll 3Y 1.. Fl't.)l'll":t¡- ':ll1,1T:'.lrls X + Y = tJ.lND X * Y = V.r AflD SfLVI\G. a 3o .)L1=l ll(?l-lLr?ll 1) 'ì[l=.1 -r¡ (2?-rrl r?1ì 4L .ìL3"=.1 1:(z.,-ll,21l 4? :ì1.¿r= 4ìD{ l',,--ìLr21) !i1 lr-1= I lLì { l7-fìL,21} 4!: 1l:lLI+rL2+f)L3+i.L¿'+DLt-(f l +ll2+rì? Ì +1, V= ( lt- I + 1 )::. {i)L2+l }:ì. ( lll-l+l } * ( DL4+l li< { nL5+1) 4\ \r=v/((llrl ) '. { I 2 + I ) :1r ( :) I + I } }-U-1 +1 V=Sîìi(.I+(:J':r(J-rriivl l +:1. " Ð5=l )-\.t t /7- 4.1 [ tr ( ,] I ..L ': . ¡ ri ) C:r T I 14 ¡'ra ''- !'= I )+ \J \ / ? a lir: ir:x,..t-t :ì:Ll.jrtGS I : ¡À:'_ 1; f HqSr c,LdSS!5, AccJRtrI!.,1û 4s. a iî t::1! (0'112'f rll'I.f,) l!ì : (cr1.2t5,12,131 !? : t0rlr2r':'r lâr2cl 4r! tv[ll f']r..] 4RFa' lF TIqFS. C ì:r: ì.itiJt tnFt tCri CLtS,S THIS t-,tX.A0 B:Ltt.lGS T3. a

È.2 C C CC ¡PJ:! TI.i: IìLUCK U¡SI3!'I O3TÀINf:D F R:] I'1 TIIE H¡XADS. . ,: SI,J;-ii\f lL jLY j'll',!P JT e TH: I¡ICID!:llC: ì1 ATRIX Fll Tllf DESIGN. : Tl-': I lCtll-jlC; ;1 ¡T;lIX lS TrrE,lRl:I ICALLY SYr4;'4:TqIC. C ' )i) 3') 4=Ir:¡í> I'i ÀT? ï { ( il r'l } = I !)l lL ¡at.=l,5.i lF (il .;r.j.Br.r-i I GJ ic 31 )íl J? I=l'.j Di l2 J=l'¡ IF (iilX':it(ti r ll. jù.1-ix¡.t(!-ìCH'Jl l G0 TC 3l 32 Ct'.1 l t'r!: '1 4T {l { (Ll r:l-')=l , ll c;:\il'jL-ì ' 1. 3 Ù aî-:'|f ll: ?rJ\arl' t5c il 2 SLICK DfSICN ilijL T0 lì. A. KIi{GSL:Y A\D R. G. STAr'l *T: \ | . . Flli'i:ll 441" '/'|til\¡llX '¡4',41, FC ì'1.ìI (5llll P'ìl'l f Ëlf'^iil 3,1 _:co Flì.tìI{r 1i l l:i: ;rll -:|.r.!

1l'jî!-.Y

A.3 $ .113 rl¡r'[\/ tr--ìC1ì r ''.)l¡l¡:l¡] r òl:liXT c c rt-l IS t:'.Gq¡v:¡ = ::rPLrrs TFÉ Ir\cIDE¡tc! C ',l1,T?, 1;.í Frlti lK36 FPCi4 rF; D'IUBL:-SI X;S c 1:ra!i; f l-:j'Lil{:s rj:,j .1 ctJBIc suqFAcr I l:4PLlr:lr lflT'.:lil  (A-Z) - ( 'ì r' ( 2 LÀjÎ ^: iì n D r r z ) , F L ,1G 6 ) , L I \tE Í I 5 ) 3 I !ll:í;::-ì'r-? lr', { 3é'},61/l?9t*C/ 4 Cfrrl (ì(l'Jlr.J=lrl2lll¡2¡3¡4t5'¡6¡TrBr9rlOtll¡12-/ r, , c ( I , J ) = I itO ( I , J ):r. ( 1r _,r I t10 I L J ) l /2+.4AX0 { I, J l+ó C C C,:,1e J- j :l-' 2C l:ltlBLË-SIX!S ÐF .TYPE I C ( ìY: 7)C I ,lq= +I -.. ì1 I .J=.'3r'-; l0 Kll= J+! Ll 1:ì l K=K:1 ,4 17_ ll 2 V=1'!' 13 2_ t:L )=C i:L ^G(Vr l+ '1Jì{ ):l , I5 r-Lr.;(J)=l Ió .1.\î(( )=l l" 't=-1 l:Ì 3 'i=\i+l 1? 2I .'r :;=r,¡1 2?- I F(FL\,-1 (^, ).:::l .l )C,*¡ T't L ?1 .- ?+ ,t=\1. 1 2\ 2i 'l(lx'l)=l ?-7 l(iìX'21=J 2? r(lXr3l=( 2t I(DX¡'+I=C(¡ iit) 30 l(lX'i)=C{1".1 ) 3t l(9Xr r )=;(L ) VZ ì(ìX'7)=C(J'()'11 13 ,)f r)x,_rl=c(.lrKì 3+ ) ( iX ? )=C ( I J ì V5 ')(rx,la)=L+:r' ' 15 l(l{'1.1 }='r+5 3E I 'ìX=)X+l

A.4 c c a1t4)') 1-i iH,.- I5 'r'lrj3L:-SIX:S 1F TypF 2 c 3o ')î i l=l'f +ù Jl=l+I +l DC 1: J=Jllró 42 lîl 7 V=lrí: 43 7 FLA;(V)=O 44 )=I Lj FLli(J)--l=Llì(i +4 l= J +7 c i.l=Àl+l /+E IF(Fi_r.;(r ).:î.1)c-ì Ti:ì e, i.l 4r) K = 5 rl -r ti= \r+ I 5l JFI jL\/".( i)..i).Il5î Tt o 5? [ ='l 5, l0 ¡r=l\+1 54 Ir(FLil(lìì.l1,1tcr rl 10 " t 5 './: rl 55 lt ¡.=":+l 57 IF(=L1c(..¡ )'.:'t.IlG0 rf 1t 5 3 l-,r ( DX r I )= I 5! I ( )X r ? ) = J +¡¡ 50 "lrXr'i)=:{J,() 5I .-ì ( I X r '. ) = i ( .J r L ) -r i,7 Î ( { 5 ) = l { i 1' } ( ' ,,, ' r:7 l ì x , ) = C ( .J , I l 64 î{rYr")=J 65 D(l{,l}=J+ó 36 ì(Iì(,I)=C(II() ir7 I { I.( I I ) = C { T r L ) ól I(IX'11)=C(L'a)' 6':r D(¡X't2)=i{1,¡l) 1) i rx=lx+l

4.5 .¡.1OiJT:- TI.I: T.\CIDÎ}ICE T1ÀTRIX 7l lñ t2 r=1r3í 7¿ .Jß=I+ I 77 )1 t2 J=J¡]r3ó 74 (T=l 15 i).i ll K=lr6 7h LL=Dflr() 77 -,/ tr L-!to 't3 l-l I F ( l-1. .:C. fì (.,'L)IKT=KT+1 7o IF((-.r'l:.1) Gil To t2 s) ¡.r ¡ ( ï r J )=l 3T '1 1( 1, Ï )=l ?.2 17_ cl ¡.ll i r:J - .? Þr,^'' -? i,r35 15 6 tllSCK DçSf G¡t DUE Tl R,. A. KII,IGSLEY AI'!ì R. G. SIA\ *TT\l I aL Pr.i^iiÌl 1i0r'1,'l i'lî.1 iÎ (3óI I ) ¡í p r I'lr 121 J7 I l12t al-ì'1 iÎ(rlr) iìl lr)C 1J. ,: i .'

A.6 qJlß ; :{¡rTr iV .b'lCia, lCWJrif ,\,îlXT .C C TillS ìlUIl'ì : Ct.,,tDlTi:S Tt-î DFSICÀl FlK¿r5 . c f ¡ri-.) îr.i Ttr:.: ïi ITÀllc¡¡:,r PLA\¡ES C T1 1 CUìIC ;,'I,FdC: I C \4!iT IS 4, F'J'ICT]I:I] I\,II I CH CCI,fPLITES tr::.trì T I-i: I¡\C¡DE¡¡CE I4ATRIX C IH: i.rtLI.lçS lrt A CU.JIC SUF,FAC:

I I rlT -ì;; -ì FUi..CT I:Àj f"î!ËT(x,y! '2 I'/rLICl- Trlr:3!.t (¡,-2, ) | \r.:,^,!,ì l(lrt?l/r*lr+*?¡1*3¡2r¿4¡5t2t1¡4t5tt¡3¡4t5r6r+t5t5t5t5¡6/ 4 .-r =:.r 1', ,r*rt, 5 3=r¡4it J(Xry) i. :-L 1 .' IF:(.:.1-l-.ì. r;ì.r.r.:1.i.¡ir.rl.GT.27)Ci:TUt\ I .r: iì_- r a rII"L:.r.)ir!.rirl\ lc r( ì.îT.1¿)i;' t) l:l I1 i ( 't. C-. / ) r:r::.t-i¡È ^l l¿ tr ( a- i..jrj.f. ) ì':T jil¡l .13 4a ll=t 14 r. î -r', ì.j t5 L3 ìt=-,.(c_tZ.tì l:, r¡?=riï-L?t?l L7 iF(1.lr.t?)rjr r-l 12 lP ^ìI=I+ liji(1:-t i-,,) 19 I F ( 11..]t .iì I .'ìì. 11.:0. ??lt¡ÍiT=I

?\) r¡ T rì. 1 2I 12 ¡il=i{,1-12r1) 2? :.?=llt-lZç2t )1 !:r:-.-t ?^: IF(1l..f.l.a1.lrì.11.fi).p2.f]¿..12.!Q.Bl..)X..lZ.Fe.82!MrcT=t ll¡ :ì:.r tì!! 25 ;l1' a 1F; r:!.J\1TI,l!! c c,-)t,i ^.,1S:1 I11.ì Ä ÌABL= T ÊCR SpEED a. 21 ilFLIaIT I¡tI:G¡r (1.-Z) ?-l I'\1T::Sì5 Pt. ¡^t: ( 41¡11¡r(21 t21 | ?-] l \rr,-:G:airZ t!(4l t15l /?_OZS*O/ 3rJ lrtT:l:Ì LIrt:(12) 3L lt I i=tr2ó 3? Jn=I +t 13 ).ì | J.:.J!,2" 31' K=V::::r(1r,,) 3; T( L Jt=( 3h I T(Jrt)=K

4.7 c 1-a.4)!1.:: rtii 45 TF IT{r;Gf:f t- pL^NES 77 ^r! r 3? lÌ? I=t,2c 3¡ .JI=I+I L) )f 2 J=.J!..\ ,2,) 1L lF(I..t:.rtl,Jl )Gl T1 ? 4?- (?= 43 1] 1'+l K-K? t ?1 44 ( .r:) IF t. "ìi.r (I, K ).;1\D.1.:O.T {J, K } } Gr:l +5 3 ¡¡'"¡r¡r¡t;" 4 +6 G'l rl 2 47 /.¡ ÞL^ j,j(\i ,l)=l -'* "l ¡Ll'l-:(\r?)=., La r)L t l.: (r,r lì=( ^- ) "i= 'r*t 5l 'a? ac'lTr lu._ . .r..!Þ(jil rt.l ; I,tC!Di 1jc: v¡.TqIX . !fì¡ rH : ll_ s I crl c i?. . { r):r L? l=1,4L ij ,!=r +1. 1+ tl 'r 1:l ,J=.r.., -ì rlri 5:i ,ì l+ K=lrl ,.' l,v=or^\.:{Ir() .7 l:r t'* qj L=1,?, !Ë{, !...r).pL.r.!_(J,Ll)Çt: -t Zl !r t4 :t.tTI.!U: .J\) :rt I I L¿ 6I )1 \'llrJ)=I ^..) .r(_r rl)-l 5I IZ i.,.1:Ì.l:lì pr.ll'il:ti t,.\ ir.t L2 3 BLICK DFSIG¡r r,.U! TO R. KI'jG-çLEy :!T-¡\l r {. ârt, R. G. STÀi.t f.; P,Jìr:'1 4C,¡! Èl?.1 55 40 ir (45.i t ) 6i )t,I'!i 2?2?.) 5i 222j3 .)P 4lî ( 1r ) lì tr..c sì:" ' 7 ) ':lll Sr-l-oY

4.8 . 1¡Îli.l gJ.l3 lr^TF iV nCG l? r ARt.t;r.jOçxT L : TrtlS'ìltJTI l: Cl:rÞUI':S A^: I\CID:ttC€ \.t4TR.lX Ê1;'I C X 1I 'I:?:IICT S1; GIVEI.I IN .A GROUP C ',¡Jil IC,l IS Tl-; lIìrCT SUM 1F 3 CYCLIC GR!'JPS a I TI.,ÊLi:IT I¡JT:GE,¿ (Ë-¿) 2 I rlT:î:ìt¡2 :¡(/)0r50) 3 Irt I:G:q tAl?,1t,t8( 2C),DC.(20) 4 a IA? iiT:e FiR *80 5 173 C: A,) +2,H0?rVrKrL ¡: ( õ i? I ? 4 Å T A 9 0 ' ï 1 , I I 3 ) 1 IE{V,.:).1}C:) Ti lO a ptJ lt l e3 ir¡ '.) 3 P?l.lT i 3.rt-lîì lt F.r' 4¡.; { l3c } ll ^3 lil ?î.l lc]=lrv 12 ln-ì^) TV=1rV I-l irr. '¡(lV,l:l)=1 .r ., ,,i I+ ? l { ) , , yti r I l5 ',¡a= i-)1:'1 1rl Lt zir'1) r f 'r {l )rCB (I } ( 1l r I=1rK ) t? ii :-:ì4ir (i1 ilt) 'DC 13 ^i {=.) I I \P'=l )t \1= 2l ì-l li Iî.=lrV 72, l"l tì i=1,,< 23 Xi= I t.r( i,j.i+i) i ( I ).u^) 2t xl = a'll (l!1+.ì:] ( I Ì,:lij) ?i XC='¡ ì.-ì ( ]'.il+DC ( I ), MC ) I V= 1r yi r. ¡a,: Xl+ Yìai.X¡ 27 I?_ '.1 (1,/,te)=I 24 .\1.=\t:+1 2a I F ( .l;. L T.',:C ) G:l T:l 13 V) 'lC=J 31 ri3:r13r1 3? I; (-.r'l .LT. rir) Gl To t3 vi \iiì='J ' 3L I .'l = J.il I it 1 3 cî f{ r I'l r_.,r[ ' vi Dl ll :v=trv ( v g 37 il pu\lliJ 4 0 r ( I V , I ) , I lì= I ç V ! 33 4at Êa)a.,l ,iT ( 30 t I I 3? ]f Gl iì 1.1 j 4C VC P!ì l^ri +4 4l 44 Êr'11.4:T ( i 1r ) 42 ST'ID r !.1 .: jrì g:\]TPY ;12 3 .rL -.Cq n:SIì"¡ llj: i- ,'K. T^(:|'CHJ , 15 5 'rL lîl'. Diil';t liU! TC K. TAK:UCHI

4.9 9JilP, l,,lAiF I V ÞC::1,\nt^l .1R¡t.t1flixT C -:HIS L )qICi.¿¡J,I CO"IVËRTS A T|,,lO-CLASS ASSOCIATION 91H':.,!: IIITTJ THF I I,!C I DE¡ICF '4ATR IX FOR 4' (+:t' 12r3l 1I Íll,r 8Y fHE r,4 f TH:ID 3F SHRI K,J ¡¡JIJT A\O SI¡IGH C I IÀlïrG:l+2 '4145¡!t5l rÅrBrCrDrF 2 CHålilTFR*80 l':Dì 3 ! C:A) +t,Hr3. + +1 -irr4tT(!101 5 lF(H.)?..iC.r I l c0 T1 30 6 oOl,if 4l,t'"0 7 I ìJNt H ..r1rl-l? B ]. 3' t-=It1) ? îi ?3.J=1,45 l¡ 33 ^¡(IrJ|=0 Il n'. L2 l=L7-? l2 q!4l,1rRrC,,_lri 13 :!{4rì)=l 14 ''1 lA.l)=l 15 u(j\,))=l 15 v(4,:)=l 17 ¡4(1r,1 )=I 13 v(9,il=ì. l,:l :.¡ { lr}}=l q 20 { :i r i ) = I 2I \'1 (Cr'1 l=1. 2? 'rf i,:tl=l 23 4(ir¡)=l 7/:. !{l^^':!=l ?5 '4(l'.'.)=l 75 !(lr,l)=1 ?7 a ( i r i l = l r¡ ?, f ) , ì ) = I 2) '4 { ¡ i } - 1 ' l0 v ( i r I l = I 3l v{i,C)=l 32 12 v(.ir)l=l 3) Pr,\:'l +0 r t'l V4 +O F,:ì:liI (.i5l tl 35 6l il I tó 30 PRIIT 4¿r 37 /+4 Ê11.1 iI(rlr) 33 It SflP 71 : \,1

$1\ITDY L23 ?L:ì1( ]:JIG\I I^lU: T.I S:IR I KHdI.ID: AND SiI\IGH lzl lL'lcK 'ltsli;'l flur T"l R. c. B1s¡

A.10 - ç¡:e';;;;ir RFGER,¡rcr¡iAplr,¡icFXT c C THIS PI'GqAt4 CC'4PUTES TI.F INCTDf¡ICE c TATTIX FC.{ 5 (35,t5,5} BL0CK DESIGN c FR!'4 A ( 5,ó) DIFFE?[r¡6[ 551 1 IrvFLl:IT Il,:rÈCER (Á-Z) 2 lÂtT:G-=ì*:2 tl|75,361 3 INïEG:C D¡(20}IDB(20} 4 alldD. l¡CI9R*ç0 HDp 5 | i.EAl) 4C, t-¡DR ó IF(TD]..ÍC.r l, Gû TC 30 7 PìTIT 4OIHTA 3 4C FtR,4ir(130) 9 ptJ\JC.-l 40rHDì ( l0 R, | À') 4 1 ! A ( I ) , D R ( I ) ¡ I = 1 , 1 5 t 11 4t FÎCYXT (3011)' 12 N¡.=0 13 !B=C 14 Dl 13 IB=tr36 15 Dil 12 I=trl5 16 X¡ì=',i lD{fj,\+t n( i }, ó t t7 xE=,.111( ¡jP+DE ( T ! ,6 ) l8 IV=1+X3+6*Xâ 19 12 ¡a(IV,I?l=1 20 l.lB =:.13 + I 2I IÉ(5¡î.1r.ó) GO rD 13 22 ¡i1=) 2a t¡ =\t:,+I 24 li cî.t II.iu': 25 DufiiiJ 4B r.,l 2ê 1+9 FllR.1 ìr ( 3ó I I ) 27 Gt! Tl i 28 l0 Drll\T 50 29 50 FIRlllT(rl') 30 S13P l1 :Nt $tt,¡TrìY t5 ó SLtCK t):SIl\ t)uã ril p. K. ^4ENCr,.t L5 ó 3LilcK DCSIG\l DU: T:l l^i. ELACKT.t=L{l!R,

A.11 c c THJS C']UTI¡]i: CC'12UTIS THl II.IC tO:\IC! MATR,I X I FOq TH! C:STG\ JW36 L I '4( lNi:G1c,*Z 3é ' l5l I L2t6*A/ ? cr'lA? ii T::rì z( L2rL2l rXX 3 e ::,ti) 41 , z 4 F11'{¡,T ( 12¡1' 5 PÊI\I 4t tZ .]r.'4 6 4l F AT ( IX, t2At) 1 lC 1 I=1r12 fB=3.:r{I-l}+t t .)iì I .J=1r12 l0 ,q=1: ( J-l )+I l1 XX=Z ( J r I ) L2 IE (XX.¡iE"rItl G0 T,-t 2 13 14(I3'J3)=L l4 ¡4(I3+lrJß+!)=l t5 ,4(lq+2rJB+21=1 T5 GC T'I I t7 IF {XX.lt lj.rL'} Gn To 3 t,t '1 (IqrJ9+1)=l 19 ,4(1q+trJ3+2)=l 20 v( I3+2 rJf, ) =l ?l Gi r-ì I r,' iì I f.l Ir ( )1 X . . I ) cl T,¡ 4 2-1 '.4(I3,J1+21=l 24 v(13+lrJS)=l ?,t :i( Il+Z.J:j+ll=l -t'",- ?_6 3'l I lF (xx.\t._:.rJr ) cr l-rj 5 2l I I = L .r1'I ?.') JT=J i+2 3) )î 5 IX=IcrIT 3t r)rl 5 JX=.JÈ'JT 32 '¡{lXrJX}=1 G' Tî I a+ IF (XX" iû. r r' G3 lO 1 i5 l:l'l-r r -çL)¡lr r I r,Jr XX 3t ST,'.)D 37 CIiITI:.IUÌ

A.12 33 PtJl.lCil 36 lir ó BLNCK DESIGN DUE TO J. t,lÀLLISI 3l Pt,.ic rJ" 4 2, !'l 40 42 Fnc'l^Í (36I1) 4l fr? l.'l T 43 42 47 Filrì¡,1åï(rIr) 43 STlD +4 TÀID (!l.r-f-iv JiII'iITIII JLLLTf:4r'!JII I'I-'MLLLÏ T i L JJiLr.'I vL I.J JLNI'ILi LJJ '.ITLLT'1 '4I TL JJI LIJ .lvL I J JL:r j .41IVJJ IL I I L:1 I IIL .I,J i LTI I4LI J .J IVILL]"-1JJ

a.13 C T'q1GP.A\4 c THIS CC¡,lPUTES TI.E INCISENCE !IAf RI X c FCa i,1= BLljcK DiS I GN JW45 c I I \r:!:q*2 ,. 145r451 /202.5\\O/ jiC 2 C!l A R. T:l F Z(15 l.XX! ZZ.*15 3 :e Ul V \Ltrl\lCE lZ I + I c=-2 'LZ 5 Ð1 I !=lrl5 + 5 I B= I3 -? 1 qz.^)tzz 8 PF. I\Ii, Z Z I Je=-2 L0 ìn I .J=l'15 t1 Jq=Jl+3 L2 XX=Z (.J l t¡ I F { XX..I'. I I I GCTOz L4 ill(iarJrì)=l ' 15 '4(IB+1,J8+ll=l l5 ( I B+ 2 r J B +Z ) = I t1 G!",t T,:l I 13 IF(XX.\jF.rL')Gn TO 3 l9 14(il,.,t+l l=1 20 u(Ii+lr.l1+2)=l 2l l'4(I3+2rJ3)=l 22 ;1 1't I 2V I F( XX..tl. r l'r I ci_rT04 ?_4 '¡{ IRrJs+2 } =l 2; :4lIì+IrJ3)=I 26 '4( I ì+2 I I =l 27 a¡ 11 r'Jq+ 2. lcrYY i.t: r ta tÊn Tn r 2? IT=Ic,+2 30 JT=Jt+2 3l r)t 5 IX=lErIT v2 ¡l1l 5.JX=JtsrJT 3a '¡ ( I I r JX l = L 3+ ûll ï1 I 35 IF(XX.50.' rlG0 To I PA. i ..li, | ÍRrlcp,, r T rJrxX 37 ST'JP 33 CCI.JTiTU:

A.14 c C:f,lrl .l:..,^T¡CI 'lF l-' ÞRCGqAt4 'l'.lCID:\C: T -= TO C0t"!pUTÈ C TH: l.rA IR. IX FUR J h45 3a PIJi.I:H, t¿r5 I2 3 8LlCK I]ESIGN DUE TC J. i^IALLIS' > lJt',î: /" 43 | 'r , "1 41 î4 F:.ìrì-(45ÌI) 42 D:r I 'li 4 ) 4a c'l"1iI(rl') .j T.l t +5 : "l) si:riïQY itlt!IlI LLL /¡.¡.¡l I vr,i,.rL[.L I I J f t_'''I i.'t I I J L"t¡v¡;1¡'.' Jr.rI LL 1"1 I L,.' .rïl\11., l''lL !T ¡,1L I J T ",1 LL¡.1 J i.1L 'L IL.1 l,4L J .ïLjliI4L J f.ÌrJLIr.!L J IIII ]L.,'J IILLL ILI: J Ll'' ¡i¡ It.:.r .J

a. t5 SJiIB a rHlt DqfGea14 cc."lDttrËs rt-F INc IDEÀtcE MATRIX FOq, =S36 I IvFLïlIl f ¡jT:C !:c {A-Z} z I¡'T:G:i'r2 tt (V5 t35l / 129 b+O / 3 ïNi T:çl). A ( 9) /l r l*0 r1,l¡3*O/ I NT:G:? 3 (9 )/1r Jr0r I rCr0' lr0r0/ i INiic:l alll /| r Jr 1r4*Cr 1r0l â I\1':G.-q D{ç }/t t I | 6+0tL/ 7 DR I'IT 1 3t Fiiì^4AT( I I t ) .l lñ l) I¡.=Ir? il I B=I1+? 1t IC=l i+11 l2 ll=!i¡+17 l3 lÎr 12 J¡=1,¡ J3=ii+? l5 JC=Ji+19 ló J1=J \+27 l7 D= 1+11 )l(c+J,\-Ilr9) p l:, ¡= r ( ) t9 '"1(I¡,,Jr)=l-X 20 rl( Iì r..l 3 )=X ::L '1(^¡(ICr.Ji)=X I )'.1 "1 )=X 23 X= 4 ( c ) 24 "'! { I.1 'Je }=X ,.1 ( I l'J 1ì=X

2- !.- ( "l I : ' J î ) = I -X 21 'r ( I ) JC ) =X 29 X=C(2!' 2') 14( ï.1 J:)=X 30 1(iSrJrl)=X' 3l '4(IirJil=X 32 1{ I I r J I ) = I -X 3V X=D ( 2) 3t+ '¡ ( l l r J .r )=X 3) v ( I']r.l l )=l-X 35 l(IlrJ'l)=X v7 l2 r¡(ll'JÂ)=X t8 PrJtjSrr,35 t5 5 ELOCK DËSIGN DIJE T0 t:. SDgNCçl 79 D:l lr* I=1r3b +0 1+ PU\jCll 41r (i"!( I r.1 ¡'¡=1r.3ó l 4L á I FCìI.1I (36iÌ ) 42 rC s1!t +v E'r Ð $:\li?Y

A.16 SJ JÌ h 4 TF I V .;iG ì r \,1/,jÂ.1!,, , NC! XT, N0CHiCK : - C ïF: F l[ Lai.,r Tr]c i i Tt-¡ vAIN IICUTIN{: F!ì . L ISTI'1i SLiJCK D:SICi\S, CDI,/PLTI\G C I\jT:ì;]CTIC..j ..¡U.,I3FiiS, AhD ccMPUTI]\G c 5f.,I îrt l,:i?'raL Fl{F c I Ii\/PL;lIT Iùtî:cIì(Â-Z' 2 I ¡.jTiG:tì y{t+tt4l/4t)5*o/ 3 Î.!ÁL'*l P(t¿) + Cii.:ìì111.:r FL,X;,'10i.ì+42,F,.4Tf!ZE(l),FFF{5},FBF*2 :iìL'j: (Ë..1T 5 :,JL I ,/ '.1_ , FF F ), ( FFF ( 5 ), F tÊ l :: ','! t r 5 Dá.Tl J / I I x , I I ( r ì r r r I I r I I : . I , I 5 , r I .tll./ 1 I f.Ì:î j-ì Kr ( 10 ) I i: ¡r_r { I,4I ) V,K r I,FLAGrr.tDR '] 4I Ft-R4¿l(lIi,.\Itia?l lJ ^rîf I:(2,+rr)rll? 11 Plr J ,ll L,c r|Dr.l 12 4) riìÌ4,ii(¡ I t ri\á2) I I I f: (?,i?l,t ¡K ¡L I!+ P:lJl'.1Þ. 42,1/,K,L 1' ?-tl li, | -,-.i1.;=r r FL.\G li '+? :1:ì¡if (, :ll'l-F ÞììA!:f SrìS y=rrI2r'r K=t,12¡r r LAtltsDA=,tlZ/l 17 Jl I l=l,v l'Ì I 'ìinl {l rrr I }( v( I rJ )r J=lr V } 19 +i Ètliti{a4I1l ¿\) -,rLL .'ì¡.;'i;,v,:() 2L CaLL trCK.(tu'P,V) 22 CALL T,llpLii(Pr'/rKrLrKIrEPRl ).) IF(lt.r.r::.û) silP 2+ LL=Ltl I 25 rrR I I-: { 2 ) TH: IiìI PL5 STqUCTURÊ 0F TH: DËS IGi{ ISI 2i, À I T i { F ô"r F , 4 7 ) L L 21 .î? FaPl.ìT(t2)^, ( ( 2,3 tJiì I I: 2tt: 1T I I-l ,KT { I ) r I =l r LL } ( 2-9 i F F L 1 G . : i,) . ' S ' ) G.l Tl 30 3ù cÂLL ïa.1\! ( .t,V,;LlG) 31 iFJ:¡1^.il.rSr) G.l T! 3C j? CALL Ð.,rCK(ùrprV) 3l CÄLL T-r I F L :: S ( Ð ,/ K K R v+ IF(iiìì.r.1:.c¡sÏ¡i,, r I L I T I Ê R I 'd ( 15 C I T i 2 r * ) ' ,\ 'Ii) ÏIiÈ Tì]PL= STÊUCTUR: í]F THS I]UAL D¿SIG¡I IS' 3¡ t'ì IT;( 2 r F.!T ) I I -LKT (I l r I=l r LL ) 3l Gfl Tl ll ': 3j lJ 4 ì I I ( 2 , ,', ) ' ,):3 I 3.¡ IS (STRCNGLyt SãLF-DúALl 39 Jl iLi53=1 4J pql l;=7 +1 3^LL i.1 lTtlÂ{irrVrtL,':SS,pRIMË} r,^l ( I 4?_ rì I T: 7 , * ) 51.1 IT.l CLiss=rrcLÂsS p:ìI.Jïr r 43 t.¡ ì Í ¡, .I = I , n: i :: ¿.4 STi!) ¿''i : .ii)

A. 17 r, ( ( R R 45 S U:l lrJîl\a T ? l P L i S P r V r K r L r C r Ê l C C Ti]IS ]5 TI': CCLTI¡iã !^IHICH V¡LIIATES TI-É C IllcIlì:"iC: r/AIR. IXr A¡lÐ CCvPL¡SS THf: Rtjl"l C I llî ji i -'iT I if.l ilUTt]:tìS C 47 Ii\/r)LICIf lllT:Gaq (,1-Z) 4ïJ P.!lL*l ?lö41t,\r JtCC'SA 4) INT:G:1 Ct10) 5ù cl:.t.\ 11 t/ t, it,cc, N f t :r¡l-u ^/ 5z 9D3I=l l0 5) I C{I}=r:) ' i4 V1=t/-1 ,5 vz-v-?, 55 D --ì 1 i=l'V2 57 J=0 53 1=S\='r(l ) 5) cìLL ìlT3 á0 1F( l.la.K<) Gl TC 30 ôI IX=l+l b2 ñ1 I J=lXrvI :r4 -,=''tJl bq Cl=Så 55 C¡LL D¡lD 66 IF(,1.'l:.1) CJ Tl 30 tJ I ./!._ r ó,J JX=J+i ô'l DC 1K=JXrV ?ü B=P{K) 7L C ALL Ì1I']) 72 ir = i'.1+ 1 73 i c(\j)=c(¡.r )+l 7t¡ G:l Ïll ll 75 30 !i.ìR=I00*I+J 76 3l PR I.lT, rLRR=r,¡Â1 77 RET'J?II 7'è i¡Jl

4.18 7) sL3?luTlrii c¡¡D c t THI S R'lÜT Il'¡t CC'1PUT!:S Tl-E INl''lER PRTDUCT C 0F T!i'l FC¡ìS 0F THI I)lCICf ¡JCE vÂTRIX' USt¡lG C ßT,CL:,1¡'¡ (3Iî STRiTIG) OPERATIONS c 8C :c,.ilVrL!\c¡(z'L) 3 D N 81 CCl"lC'l/tal\ I A ' r C ' r 82 X=A'lD(Cr¡.) 'Y 83 Y=â:'lD(irB) B4 ENTRY ìITS 85 lt=o E6 Z=X 87 t lF(1.:c.0)cl Tc 2 88 N =:'.1+ I 8j Z=.r.ID{L'L-I) 90 GrJ Tl I al ) Ì=\ 92 3 IF(1.¡:1 .0)rãÏrJRrl .13 Ì'l = l.l + I t4 Z=A.lDtLrL-l) 95 G'l r:l 3 Iô l:hlD

S7 SLili.lJT]ii:_ 'f .ì\l.ts(¡1 ,V,F¡ C C TIJIS r:UTIi.:: TilåNSPa.lSlS THE INCIDF¡tC! C ¡14ïìIX, :riD SItvlJLÌA;i:0USLy CIiECKS Ti.J C S[5 IF Tt-:: .1¡TPIX IS SYrvtvÉTFIC { Ir\ !,tHI CH C CAS:, TF: D;S IG'J I S STRITIGLY SELF-DUAL c 93 I¡IT:G:ì v r!.11ô4,541 ?9 CHAì¡iCT:Ê F IOIJ PR I i.J T, ' STAF T I i\G TT-I TRANSPCSS I I0I F=rSt LOz ¡/iu = V_ i loj ¡3 I 1=l¡llv t0+ I.J=I+I lOt Dí-ì I J=\l .v 105 "1U=4{l'J) 107 '/L='4 (J r i ) IOE TFI.1 J.:i.YL}GC TÛ I 10? :'1 (I'J)=ÍL 110 ^l{Jrl}=¡U lll F=rrlr ll2 I CC¡tIIrliiã lt 3 RãT'Jqii TI4 iND

A. 19 llt Suì:t-.ltjTliti. .;MITHå(lr',V,N,pI, a t THIS RiUTIri¿ CALCiJLATT S TFF S.,i ITtl c .trlÈ,1i1 f-i_Ê .1 lF rHe I tctDËuc! v,AT¡IX a 1l'; i:vFL IC I I I¡ îÍG Lr ( A-Z l tl7. IriT:3_.ì :t,tt-:t,a4l ¡À (6/r) Il8 1 t.,r :l j=t,V Il-) rìll i J=l,V 12il :v''¡='1 (l'J)+2 l.2I Gri I.l (2t1)Zltv'\1 12¿ ) (.t \it tr_i 123 ¡,1 i I=l,V L?4 r'-. ,+ l:L\/ L25 Ir(:)..l_.i/"iDi¡{LJ),p}} G,.l T0 30 12'¿ r Ciiìrl l,_: I27 r:i,?.i 123 3C i,=c 12i ?': j J: r llJ Z )¡: 5 JJ=I,V lll i ir(J.J)='i (Ir.,J) 112 J= j+I I.?l IX=l 134 l'l ¡ II=lrV 135 I.(i.-,i1 Ì3 ó Lis L=v(Ì1,.i)";1)GJ Ê 'r r; 137 I ( , n . . ., ) L = - L ll.J JX=I 13J ìt 7 .J J=l,V l4ù IF(.Ji.:.1.J) G,l T:l ? (l (? I4l '.1 JX )=)r (l l rJJ )+¡,:1¡33 ¡ L42 .,X= JX+ I Lq3 / CCrtItiiLi: 14+ lX=iX+L 145 5 CCiiIIliL:: 14.J V=V-l l+7 jF(v.iT.o) ûc TiJ I l+9 ,sl TtJì'.j 14) !\lj

a..20 15ù jJ:.. îi:':'t: .'-ìik (tr'r pr'/)

c lHI J I:LíIIi: Þ I.:KS TF'C INCIDE¡IC: I.,4AfRIX C l1.'T I ,Ìi'Li:,,) l (;ilT-sîÊ I¡J!.;l FCRf,l Ft-iR j¡.tr; C P;ì li;.ss ¡Y 'TRIPt_tSr C

lil ir,lirlic iÌ it.T:;i:ì (¡i-Z ) ! rt -\r._ ¡(¡-rrc¿rt ..!-,,¡.:r1 L53 i) ({r,,. } I D \;r+( z } ( ' , I lj-:t -: ii 'J I / \ L .. C . ; r )i r I Y ) , ( X ( 2 l , I Z l l5Í )r ll I=l,l"l 15: IY=J 157 I l=a li; ' ',.: I lli,; 1f ¿ l=l¡l-Z -- li¡.) IË(!(l t a)- j'). (r . 1 ) I Y= I Y + l,! r'.1 :,' t, r 1Þl,..¿ i, = t +'r. J5=a,',-v ¡:v 1óJ =?:: 1(Js_] ) L6/+ l: 3 .l =,tSt:? L)t í F (" t L: l-.J ) . ii).1 ) I /-= I ¿+u^{ l'- ¡' J lÒ 7 l. I D(i)=) I ,'r 'J 1-r iT I 'l lrl

17.) ls:JrJrrìur I.l: Þ,ì. Ii,T()l ,,/r() Trrls ì:LTt\d PiìtitTr Trì5 SlBi ¿ tìY LI iII'1c I!,: V1;ì l!Tit:S wHICH iic J ì I',.J ti1i"il itLitcf\; Tl-: Liur¡Bi:R c iT Iil: l{tl lF :ÅCil R (-Ìi,t IS Ihi :J!_,1:i( \iJ,riii,ì, .j,.liJ TH: (:Tt FR, NUMq:RS ¡..ìr rr.: v,;Ê. I_:TIi:S I\CI0Ë¡tI hIl¡1 IHôI _ìL lc\ ÞLIil l7I Iri l t I¡lItG:,ì( Å-z ) t7? Ii'iIaaì!I ¡1.{ . + r ú 1r } , L I N: ( 3 0 ) 11 t 9l I .,=1rV l¡ + L.x=C L7a D:l Z I=1'V l7ò iF(.). jî.:¡( LJ ì)C:r' TC 2 LX=L:{+l 17 :. Li\i:ILX)=I lZ) 2 ',J.) C.:\lT;.lU!_ 1 I riì. ll: (2'.iù )J r ( l. I li( I1., t=l,LXÌ 13 tr: (1X, t 4ù ì'1 i- J ? t' it , 3l11/ l5X.-3OI4 ) ! L:t 2 l;rì a,.27 " APPENDIX B

LISTINGS OF DESIGNS 3L'lCK n:SiC\ iî!,8 Tt :. SeÉNCF l'lI IH ,)¡rl \ ì:l::rìS V=36r K=t5, LAVBD'1= 6 t: . 4 7 3 9() t0 t3 ló 19 2L 26 2B ?a 36 1 .+ 5 3 ll 14 l7 20 22 21 28 29 30 2 4 5 5 9 L2 15 l8 19 2t 23 29 10 3l z 1 5 6 1 to 13 16 20 22 24 30 31 32 3' 4 6 7 8 ll. 14 l? 2L 23 25 31 32 37 5. 4 5 1 3 ,a t? 15 l8 22 24 26 t2 31 34 -li 4 q ; 4 -t8 lc 13 t6 2V 25 27 33 31 15 2 , t: I 11 14 l? 19 24 26 34 15 36 ? 3 a 7 I 12 t5 18 zo ?5 21 28 35 36 1o: 4 7, lC 14 15 lt 20 27 29 31 32 33 V4 36 1l: 2 a 3 11 15 1ó 19 ZA Zt ?_3 30 32 33 )4 35 1.22 7 5 e 12 1., t7 ZO Zl 22 29 31 3' 34 35 76 13: I i 7 13 17 19 Zt ZZ 23 28 30 72 34 35 36 L¿+t 2 \ 3I0 14 lî 2?2324 ?_\ 29 3I 33 15 t6 I5: I å 9 l0 lt lã 27 24 Zs 28 29 30 3?. 74 i6 li: L á 7 II I2 ló ) 1, 24 25 26 28 29 30 31 )V ?5 L7? z L¿ Il l7 2t 26 27 29 30 3t 32 1/+ 35 l8: 3 6 9 13 14 13 19 26 21 2a 30 3l 37 33 35 t.?: 1 3 8 12 L3 t4 15 t5 t.1 la )7 tL 2? 3! 3t* ?,4| 7 4 e 1l 11, t5 16 17 13 20 2+ 25 2,9 32 f5 ?1: I 3 5 IC t+ .IS lí: I? 18 7t 25 25 30 3V 35 z'¿| 2 \ á l0 11 15 tô tl l8 ?-2 26 27 ?3 Jl 34 Z)2 3 5 7 lC ll t? t5 L7 lB te 23 27 29 32 15 ?-4. 4 i 3 10 ìt t2 l3 t7 I8 19 20 ?+ 30 7i 36 25r 5 1 ç 10 l1 ).? 13 t4 t8 ?o 2t 25 ?-5| .i 28 tl 34 I 't 3 10 tl tz 11 L4 r.5 2L 22 26 2.q ?2 35 ?'7. ?_ I tl I? 13 14 15 tó 22 23 21 30 i3 36 2?t. L Z 9 10 12 17 20 ?I Zz 24 26 27 28 )2 33 2q. I ¿ 3 1. 1'I] t8 1? 2t 22 24 ?.5 ?7 2.4 ,33 34 30: 2 ) 4 tC 12 L4 19 ZO 22 23 25 26 30 34 35 3I: J 4 5 li 13 15 t+ 2C 21 23 24 26 27 31 15 3ó 3?: 5 5 L?, 14 ló 19 ZI 22. 24 25 27 28 ?2 36 f3: i 5' 7 13 t5 17 tç 20 Zz ?-3 25 26 28 7-9 33 34| ô 7I14L5 l8 2X2t23 24 26 Z7 29 i0 34 35: ? 3 ç l0 15 t7 1? 2L 22 24 25 21 30 31 35 3ó: I 3 9 ll tó 18 19 ZO Zz 23 25 26 3l ?2 36 TIJ-: iJ,i PL.: STPI-ICTUl: ÛF THç DES IG\ IS Rl= tl4t¡ a.2= C 'ìI= 31f3 ?3= 2l4Z R4= 39ó R5= O I'LI! 4'II ï'iJ PL: Si4UCTUR: "']F TI-: DUÀI ¡¡5IGN IS Fl= J ?1= 3+t R?= 7153 ?.3= 2142 R4= 39ó R5= R6= l S'lIrH CL1S3= tó B.l BLfCK ÐISIÇ.I IUT T.] J. WALLJS pADA'1 !,ii TH :T2ìS V=ló, K=l5r LAqßDA= 6

i 6 7 4 9 t0 tj. t6 19 22 25 28 3t 34 i a, 7 I q ll 14. L7 20 23 25 29 12 35 5 ó 1 I c 12 15 l8 2t 24 27 30 33 .36 4| 2 ' 3 1 4 9 11 14 t7 ?_I 24. 27 23 31 34 .?, 3 7 I ? 12',15 t8 19 22 25 29 32 35 6: 2 3 7 3 9 l0 13 1ó 20 23 26 30 11 3ó 2 3 4 5 6 t2 15 18 20 2t 25 28 31 34 2 3 4 i ó ri t3 t6 2t 24 27 2e 7? 35 2 3 4 5 ól1 t+ 17 t9 22 ?5 tc l3 36 10: á 3 13. 14 t5 16 l7 18 19 21 27 23 -t3 35 ¿t I 13 14 15 15 17 tg 20 ?.4 2i Z9 3t Zb l2t t 1 t3 t4 t5 L6 L1 13 2L 22 26 30 32 t4 tl: t .9 tf, lt t2 t5 17 tB 20 24 25 30 72 34 14: ? 4 9 l0 tl t? L6 l7 19 2t ?2 26 28 33 15 15: 3 5 7 t0 tt l2 16 11 t8 19 23 21 29 31 36 L5r I 5 ? li) I I t2 13 14 15 2t ?,2. ?.6 29 31 75 17: ? + -c tc 1ì L2 t3 L4 15 t9 23 27 30 3? 34 I i. r 5 7 LC tt t2 13 t4 15 20 24 25 2f 3) l5 5 . 13 l5 ll 2? 2V ?-+ ?.3 26 27 23 12 3f, 2Q? ? t 7 1l ll 19 2? 23 2+ 25 26 21 2c 13 34 ?.12 3 + 8 t7- t4 t6 22 21 24 2_5 26 27 ,lO 3t l5 5 ?t2L4 15 19 20 27 25 26 27 2s 17 3+ :> 7 10 15 l7 l9 20 2L 25 26 27 3O 3t 35 + ,e 1l l-l t8 l9 2c 21 25 26 ?7 ?3 72 3f) 5 ,a rl 13 18 1? ?-O ?_I 22 23 ?_4 30 31 t5 Zht 2 5 712 l1r 16 19 ?O 2L 22 23 24 28 32 3ó ?_7 2 3 + B t0 15 t? 19 20 21 22 21 24 29 33 34 2?r I 4 7 lC_ t4 18 19 24 26 3t 32 33 34 35 36 2e3 2 5 13 Lt 15 1.6 2C 22 21 3t 32 rl 14 15 t6 -3C: 3 I I 12 13 17 2L 21 25 3t 32 33 34 35 36 3l: I q 7 ll l5 ló 2t 21 25 28 29 30 34 i5 3ó 37. 2 5. I tZ t3 t7 t9 24 ?.6 28 2e 30 74 ]5 36 33: V 5 9 l0 14 18 Z0 22 27 28 29 30 34 35 36 14t I + 7 12 t3 17 2A 22 21 2g 2e 30 3i )2 33 15: 2 5 I l0 14 t8 2t 23 25 28 2e 30 31 32 33 15? a 5 9 li 15 ló 19 24 26 2q 2s 30 ll 32 33 TH:: TRIDL: STIUCTU9Ë li THF DFSIGtt IS a0= l¿r4 CI= 469 R2= 3og6 R3= 2232 q4= 288 R5= 0 R5= 12 DiS IG..I T.i (STq]IJGLY ) SãLF-DUAL 91! I TiI CL\.SS= r2

8.2 BLlCK DES iG\ CU! T] (. TAKEUCHI DÁ"ìA Wi Ill l:TÊaS V=lór K=l5r LAyBDA= ó

l: ?_ 3 4 5 9 13 t4 17 19 2L 24 26 32 35 2 3 4 6 to t4 15 18 20 ?_t 22 27 2e 36 2 3 4 7 11 15 t6 t7 t9 22 23 28 l0 33 2 3 4 3 t2 13 16 18 20 23 24 25 7L 34 5 ê, 7 3 ç t3 ló 17 lE 2l 2_3 21 i0 36 ?. 5 6 1 s t0 l1 14 18 19 22 24 28 3t 3i 3 5 /¿ 7 I tl t4 t5 19 20 2L 7_3 ?5 32 34 4 5 6 7 3 1.2 t5 tó L7 20 22 24t I q 26 29 i5 I0 1l t7 lJ t5 l? 20 2t 22 ?8 3l 34 2, .> 9 l0 lt t2 14 t6 t7 r8 ?2 21 25 32 7 3' lt: 3 ç tc Il t2 13 15 t8 t9 23 24 26 Ze 36 l). 4 r. a t0 ll t2 14 ló l9 20.2t g 24 21 30 33 2_ tl 13 L4 15 tó 17 2r 25 26 29 3t 33 3ó : l2 13 11 15 1ó t8 22 26 21 30 32 l3 34 4 o 13 t4 t5 tó 19 23 ?_1 2A 29 t '1 31 14 35 13 1/+ 15 16 20 2+ 25 Za 30 12 )5 3ó l t) tt It 17 t3 19 20 2L 25 28 29 30 )3 35 4 7 9 L4 L7 l9 19 I 20 22 ?5 26 30 31 34 36 it 1? i8 iç 20 23 26 27 5L 1¿ )3 J5 2_ 5 lt t6 17 1€r 19 ?o 24 21 2.8 29 12 36 4 'l 7 l0 13 L7 ?L 22 23 24 25 27 ?.9 32 33'4 3+ I 8I 1l L4 13 ?,L t-l . 22 2-i 24 26 28 29 30 34 35 Z 5 L2 t5 t9 2t 22 23 24 25 27 30 31 35 36 3 r¿ 9 16 ?) 2t 22 23 24 26 28 31 32 V7 36 I ?_) 5 1 9 L2 't L4 20 23 25 26 27 28 2c] 13 ? e 3 c l0 15 n 24 2_5 26 ?1 28 30 t4 2_7. qr.l 5 7 l0 tt l5 l8 ?t 2s 25 27 28 3i 35 qtr 6 B tt t2 t3 te 22 25 26 21 28 72 ló 4 5 â 1 lt 15 18 2.4 25 29 30 31 ?3 ¡c t 12 2 5 7 l0 t2 ìó 19 ZI 26 2e 30 3t 32 V4 3t: 2 3 7 I 9 ll ll 20 22 2-t 29 r0 31 32 35 v + 5 310L2L4 17 23 282e 30 372 I 31 32 t6 3 5 I I l0 1ó 19 22 25 29 v3 34 35 36 2 4 5 ó lC ll t3 ?_o 23 26 30 ll t4 15 16 I 3 e 7 lt l? t4 L7 24 27 31 t1 14 V5 36 ?- q 1 3 9 t2 t5 l8 2t 28 32 33 34 V5 36 T!".1! TR I DL STqI'CT!Ifì1 ,]F TFE DES IGN IS 7 'P-1= ? ll= å43 R2= 3B8S R3= 219ó R4= 324 R5= a5= I 2 å:.1ì 1'r: ìI FLE STI,I,,CILiR;: Í]F TFÊ q2= DËS IGN IS al= b48 3888 'UALq3= 219ó R4= 324 R5= c.!lT;l rr ^ lS= l4

8.3 p. aLílcK D:sIG\Jr:ìc iì,J!. t'-l K. vsNlN '¡:TH rÂq\..::f ,,t=3L|. K=[5, L.lMBDir= 6 '3 li 2 5 7 ?, 9 r1 13 L4 L7 24 25 26 21 t4 2| 3 4 t 8 9 10 t2 1+ 15 13 19 26 27 28 35 .l: I t 5 ? c 10 1l 13 15 tó 20 21 28 79 36 5 /: 3 lC 1l 12 14 16 t7 2t 28 29 l0 lt 5: I 1 I i ç tl t? 15 l7 18 22 25 29 30 12 i', I ? 4 7 ? .10 1.2 L3 ló l8 27 ?5 26 30 i3 I ç 11 13 19 q: t4 15 t7 2A 27 30 31 3? t3 5 ì I0 !2L4 1' 1ó l8 20 21 24 2-5 72 13 34 ? 11 11 13 t5 tó t7 19 21 22 26 3i 14 -?5 l0: t 3 tl t2 t4 lá t7 t8 ?o 22 23 27 34 35 33 11: ? 7 ç t2 13 15 17 t8 ?_l 23 24 28 tl ì5 36 I I to 13 1+ Ia 18 L9 2?_ ?4 29 31 1? t6 l1: ? ) tc t4 15 L7 19 20 21 21 25 26 2a 16 l1+l ,? Í ll 15 lr: l8 ?_C ?_l 22 2+ 26 21 'ìt 3l lii: 4 5 l-? t3 \t t? t9 2t 22 23 2' 27 22 12 l'.¡i 7 6 7 L.+ 11 l3 ZO ?2 27 24 ?6 28 29 33 '; :) I i3 l5 l'Ì 1? ?l 23 24 7_7 29 iC 34 1:J: 7_ a i t3 l4 1ó t_c 20 7-2 24 25 2s l0 35 l+: 6 1.9 ç ló zc 2L 21 25 2$ 27 2? 3 t ?,?. 35 ¡çI0t72t2224?6 27 7E 30 t2 33 36 ?lr 2 r lt lt 13 t9 ?.2 2) 25 21 28 29 ll 33 34 ??_2 3 l0 11 L2 13 20 ?_3 24 26 28 2.4 t0 32 14 35 21: 4 lI L2 t4 19 2t 24 ?.5 27 29 30 33 35 3ó ?/+ z 5' a, L2 t5 19 20 22 25 26 28 lC ?l lt+ 36 Z5r !. 5t? r1 14 L5 2225 77 29 3t 32 77 35 ?5: 2 t 7 t4 15 15 2i 2'l 28 30 3?. 33 14 16 27 1 r + 3 15 1ó 17 ?4 25 28 2e 3l ti v4 15 28: 2 q a I ló l7 t8 t9 26 29 3û 32 34 J5 36 2a2 a a 6 IC 1t l? 18 ?o 25 21 30 31 33 35 36 l0: I ¿t 6 lr 25 jl: 13 14 tB 2L 26 28 31 32 34 36 | 2 I 5 7 9 lt 13 lç 20 2L 2a 32 i3 ?5 i2'. 2 t 4 ó I a tZ t3 20 ?_t 22 29 3l t+ 36 331 I 3 4 5 1 9 10 1.4 2l 22 23 30 31 74 35 J/+: ? 4 i 4 3 l0 lt t5 Zz 23 24 2' 12 35 36 3¡: I 3 5 rr I ll 72 I5 19 23 24 26 3t 33 t6 162 | 7 L $ 7 lC l?- 1.1 19 20 24 21 3t 32 )4 TH:r: :'ìlPL: 3I?¡JCir]l:l ilF ïHF DiSiG\ IS q0= Ií+4 Cl= 1+53 \2= 3cJç5 ?7= 2232 R4= 2gA R5= 9jr= l? ,\:.i|,1 i : Tî IPLE S'ìIJL]'UR: 'ìF THF ]U1L D[SIG¡I iS R0= L44 ìl= +ó3 lì2= 3c9ô Rj= 2232 R4= 28g R5= R å= l? s.1 iÎÊt CLAîS= t2

8.4 ÊL]CK IlS!G'I ]J: T --I ?,. q. K II'IGSLEY A\D R. G. STANTSN r{iTH Þ111r:T.:,ìS V=36r K=15, LAVBC^= 6

7Z 2? 21 7_5 2.1 28 29 30 31 32 t3 34 35 t6 q TT 15 l.å t7 13 t1 2C 22 23 ?1 lq ?5 3t: 't .36 3 l1 1 -1 L4 tl t3 19 2L 22 24 7_S 72 37 :l . lc t) ró 18 ?_O 2t 22- 25 29 31 13 35 ? t2 l3 15 19 20 2t 22 26 .30 31 3?. 34 5 tl L? L4 t5 tó 20 zt 2, 24 ?9 30 31 76 1/, ic L?- t5 l? 19 ?t 27 25 ?g l0 1?- -3 5 q: 1 + 9 I2 1_l L6 L7 l8 ?_L 23 26 28 29 33 34 q ?. a t? T3 tó 17 19 ?o 24 25 21 3C 31 34 -l 1,1 : 2 d L2 l4 15 1? 13 20 ?.4 ?.6 27 29 32 75 .l I : Z Ò l3 15 t6 18 19 25 26 21 28 31 76 '! 12: 1 3 I 10 t7 20 2L ?5 26 27 23.31 36 x i ó 3 15 la 2l 24 26 21 7-9 7.2 V5 l4: 1 1 10 15 18 2L 24 ?5 21 30 . -17 34 6 1 1C t4 t 9 20 23 26 23 29 t3 34 l6 ': 5 fì o 13 I8 20 ?3 25 2A 7q 32 35 17: 3 7 81 r0 12 l8 l? 77 24 2? 70 3t a6 8 l0 lt t4 L6 t1 2.2 7_6 l0 31 1?_ 34 -l I f . n ?r) ii 13 t5 Ll l¿ t2 ¿"e Jt ¡) t) 7- 0.. 4 5 5.] l0 L2 15 1ó 22 24 28 32 1ì 3ó + l 67 I t2 ll 11 22 23 21 34 35 i5 2?: ?- 3 4 18 t9 20 2t 3t 32 't7 34 l5 36 23| 7 7 3' 15 t6 t7 2t 28 29 30 ,4 35 36 l 6 9 t0 t3 t4 L1 20 21 29 30 32 33 36 4 9 l1 12 I+ 16 t9 21 28 3C 31 33 15 t c lC ll r? 13 15 18 27 28 29 3l i?, t4 7 ç l0 lt t2 1.3 t4 2t 24 25 25 34 3r,, 36 3 7 8 lr ï2 15 16 20 23 25 26 32 .13 36 7_a 2 + ó 3 L0 13 15 t7 19 2) ?4 7_6 31 33 35 3Cz 5 5 1 9 t4 tó 17 13 23 24 ?.9 31 32 34 31: 4 6 lt r? 17 t8 t9 22 ?_5 ?_6 2e .3C 36 32. l 5 'I7 l0 13 t5 18 20 22 24 26 28 30 35 3 4 I t4 15 19 20 22 24 7-5 2â ?a 34 11a 2 c I I 14 15 18 2t 22 23 26 27 30 13 2 4 7 l0 13 1ó 19 2L 22 23 25 27 ?,e 32 3ô: ) 3 6 lt 72 17 20 21 22 23 24 27 28 31 TH: TE. IPL: SïFiJCTJìE CF TI.1F DiSTGN IS qJ= 0 il= i+O P.2= 4320 R3= 2160 R4= R5= P 3= l2A 1: S I lì^! lS (','lrìir¡.lLY S;1tr-DUAL S"!JTi-J CLdSS= ' t5

8.5 ìLlcK l:slc',r rlui T.l i.\!. !lLÄcKhtrLD:c t{rTH D1ì\v:1:_:qS V= Í, I K=15r LA"rFìDA= 6

:j 5 7 L2 t3 11 19 22' 25 21 31 32 :J 6 7 3 L4 18 20 23 ?6 28 12 a3 ô 8 a 13 15 2L 24 27 29 ?i 34 r¡ q t0 14 t6 l') 22 ?A 30 34 35 4 [0 1l 15 L'l 20 23 25 2? 35 36 4 '¡ tl L? t6 t8 24 25 30 31 36 ')_ c lc 11 12,11 t8 l9 27 2.5 2E ll 33 ,' ll t1 12 13 Lt¡ 20 24 26 29 32 34 ) + 11 11 L2 t+ 15 l9 2t 21 30 33 35 tc: ¿ 9 l1 t2 15 16 2_O 22 25 28 1+ 36 t5 3 I IC t2 15 t-l 2l 23 2b 2e 31 35 l. ) J 10 It 17 L9 ?.?_ ?4 2_7 3ì 32 76 I 1 1 3 l4 l5 I t8 t9 2+ 2a 2S 3t 34 .:. 1^ : ?. ll 15 ló t 18 lc 20 ?,.5 3C 1? 35 lc: I l] lí) T4 It. I 18 ?-O 2t 2-5 21 33 35 Iil II L/+ 15 r 18 2l 22 7,5 2-8 " L )4 lt t?, l1 15 !5 I,3 2?. 23 21 ?-9 32 35 1lt: -7 L2 !4 t5 ló 17 zz 24 .2s 30 73 V6 l!' 2C 21, 2!* ?ç 3Ð 11 35 .a 11 l/+ la ?_L 7_7 23 24 ?-5 25 3? 1ó 'c 11 t,; 11 7,3 ?2 23 ?4 7_6 27 3t 73 IC 12 ilr 21 21 ?_4 Z7 2E 32 34 7 l1 11 L9 20 21 2? ?4 28 29 33 35 )L. 7 3 L2 13 I3 t9 20 2t 22 27 2.a 30 V4 36 I 7 lo 13 15 19 2î 26 ?7 21 2_9 l0 jl 36 2_ I 11 1+ t6 20 2l 2i 21 28 29 _t0 31 12 I .e I? t5 17 2L 22 25 26 28 29 10 3Z 33 2 ? t0 t5 t8 22_ 23 25 25 27 29 30.33 34 ) I tI ll 17 21 24 ?, 26 27 23 3C 14 35 4 9 L2 l+ lq t9 ?_4 ?5 26 ?1 28 2e 35 36 .ìt: 1 7 11 t3 16 r,ì 2t 25 26 12 t3 34 35 36 I q 12 t4 17 20 22 26 27 31 33 14 l5 V6 2 1 3 15 t8 2L 2'l 21 ?8 vI .i2 i4 l5 36 ) 8 l0 13 1ó 2? 2+ 28 21 3t v2 31 15 16 .l!: 4 o 11 14 Ì't Il ?7 29 30 31 3? 33 14 35 I0 l2 15 t8 2C 24 25 30 31 t2 33 14 35 -1- Trli: c J FL : TqUCÏi',R i: iIF IHF D!S T3\ I S i: 4å8 .R.2= 39_c6 ?.1= 2232 R4= ?98 R5= U Íìi= 12 l::.S I G"l M Srì lr'jGLY ) . SELF-ntJAL s rl-rt cLiiS= L2

8.6 qL,!cK c:sIG.l ìU': T;t J. hALLIS r¡J câq I rH 1'1:1':RS V=-+5 r K=l2 r Lt143D¡= 3

t: 319 2225 2831 34 37 40 43 I 20 23 25 2? '?2 35 3A 41 44 3: 2. 3 ?L ?4 2'r 3D 3V 16 39 42 45 6 ZO 23 25 3J 33 )t 31 40 43 + .l \ :. 21 24 2'! 23 31 34 33 41 44 j c 19 2? ?5 ?9 3? 35 39 42 45 7 I 1 21 2q 21 ?a .12 15 a7 +o 43 1 a -?O Iô 2 2 2q 33 .-? ó 3E 4l +4 7 ?,c 27 2a 2q 31 j4 39 4?_ 45 l0 t;? 1ì 23 ?i 28 33 35 37 4t 45 I I: tl t?- 21 ?4 25 ?9 3t 3ó 33 42 13 t.l t2 zL 22 2b 30 32 34 39 40 44 l3: ll '!.1 lq 21.) 2+ 75 30 22 14 17 4t 45 1i 1; 21 27_ 26 ?g 1V 35 38 42 4V lr': ll l4 15 1>^ ?3 21 29 11 36 39 4û 44 I i; L:) I7 IJ 2t 2? ?6 29 3t 36 37 4L +5 1; l1 l8 Lr 21 ?7 3C 3?. 34 38 4 2 43 I ). l5 l7 1J 2a .24 25 2.e 33 35 39 40 4e. t 5 I 10 15 t7 lc 20 21 37 42 44 ?, 5 11 1l 18 19 21 ?L 3ts 40 45 2t: 3 4 ! 12 Lt+. 16 19 ZO ?.1 39 41 +3 ?.? | 2 5 t2 74 t5 ?.2 21 24 37 42 44 g 3 + l0 t5 L-7 22 27 24 33 40 +5 3 ç 11 13 18 22 23 24 3? 41 43 + 11 13. l8 25 ?_5 27 37 4? 44 5 I 17 14 1ó 25 26 27 33 +o 45 f 7 l0 15 I-t ?5 26 21 39 41 43 5 ? tc L4 t8 19 24 2.6 28 29 30 ? l. ) /* : li 15 1ó 20 ?-2 27 2n 29 3C lrl: 3 5 L? t_¡ t1 2t 23 ?5 23 ?9 30 ll: 3 7 tr 15 t6 t9 2.4 26 3t 12 73 VZt I a 8 12 13 11 2C 22 ?-7 3l 32 33 ti to 14 l8 2t 23 25 31 32 33 I c 12 13 tl 19 24 26 34 35 3ó 5 1 l0 11 19 2C 22 21 34 35 35 lj¡: 3 I 1l 15 1ó 2t 23 25 3+ 35 36 q 7 10 l3 tó 28 3?- 36 31 38 39 3 tt ti t7 2a 33 34 37 38 19 9 L2 15 r8 31 31 i5 37 38 3c '?9 ll Iq t7 2E 72 36 4C 4L 42 íl: I r' L? It tB 29 33 34 40 4L 42 1?-i ? 1 .-r t0 13 ló 30 31 35 40 4t 42 41? ? 5. E IZ 15 t8 23 32 3ó 4Z 44 45 t-'¡'t I 6 c l0 13 16 ?ç Z? 34 43 44 45 452 I { ? tt 14 t7 30 31 35 43 44 45 TI{I: TiIPLi: STJi.JCÏI,iq I 1F lHi: I: S IGI.J IS ¡t')= 52?J ? l= 3t0l R2= gt0 R3= 6¡ åI;!' THi T:. I ÊLI STEUCTUN: .-1Ë I[rE CU4L DESIGN IS Rl= 522J ct= ElCl r\2= StC R3= 60 SI.I] TH CL \SS= 8.7 BLIICK D:S:C.I I.J: T] K. '!"{KEUC!JI r¡il 1)lq.^r.1 TH :Il:RS V=45t K=l2 r LAyBDA= 3

t: ? {+ 5 712 1A 2430i34044 /+ ¡l li t9 23 ?6 34 36 45 v'. .4 5 9 l+ ?c ?t 27 15 )1 4I ) 5 lt t5 t5 22 2A 3t 38. 4?_ 1 5 Il 11 ..23 29 -aZ 39 r+1 2 3 1 lo t? 20 ?1 29 14 38 r+5 111 ll 1ó 24 30 357!'+L 3: 7 I t0 -l L4 l7 7_5 26 31 40 42 5 fJ i0 t5 ts 2t 21 32 t6 43 i 7 3 ? tl .la ?? ?q ,33 37 44 1l: 2 'tIL2 t I L¡t 15 19' 25 2g 35 39 43 1 13 1.7 t5 2C. 21 2s 3t 40 44 t+ lt t? 14 15 15 ?2 30 7? 36.+5 14: t ! 1 L2 L3 15 17 7_3 25 ,' 3l v1 4t I L l? 13 14 13 24 27 1+ 33 4? l t') 17 13 to 2C ?2 ?7 13 39 45 + Ii Il tç 2D 23 28 f4 4D 4t 5 t l{; Ll. 19 20 21t 2q 35 76 qz I 3 12 I/¡ 17 l9 20 25 3C 3t V7 4i ;! rj I ? r ii lô 17 I8 19 2t 26 32 18 44 3 t5 17 ?2 ?7 2_+ 25 21 35 38 44 :) ç tt. t,l ?l 27 ?,4 7_5 2'.3t 7t 45 I Lf L? lq ?t 22 24 25 29 32 40 41 z :) t3 2) 2t 2? 2t 25 3C 33 36 42 7,4 | 3 7 !4 li; 2L ?7 21 24 26 34 17 4) j :} 13 r'7 2? 2'l 28 29 43 1'7 . 30 34 40 I lc l', t3 23 26 28 2.a 30 35 36 44 7 1) t5 l9 24 76, 27 29 30 3I 31 45 2a2 3 7 ii 20'2, 26 21 28 30 32 38 41 c' I?- L(: ?.1 2t 27 28 () 29 33 39 42 31: 3 15 1ù 25 ?e z2 t3 34 15. t7 42 V2. + TC 1. 1 19 ?L 30 31 33 34 35 38 41 33: 6 L? 2C 7_2 26 3 t 3?_ 34 35 39 44 i4: t 13 1(; 27 ?1 3l z? 33 35 40 45 2 8 14 11 2{.t 29 'lI ?Z 13 34 36 4I 5 9 I1t 19 21 30 32 3"? 38 39 40 t+2 1-t '. t I 15 ?O 2i¡ 26 23 36 38 39 40 r+3 3ÊÌ: 7- ll 1l lú 2i 27 34 36 31 39 40 +4 3 6 i2 L7 21 28 ?5 36 37 38 40 45 l0 3 + j'1: 7 11 I9 22 20 31 3ó 17 38 39 4l 4 t0 t3 t,2: 20 2L 28 3?. 37 42 43 44 45 i 5 l¿.¿ ló 25 2t 3V 38 41 43 4L 45 *3 z I 1 Lt 17 2t 10 34 39 4t 42 44 45 44: ? 3 lt t:l ?? 26 .t1i 35 4A 4t 42 47 45 ,l ¡ 1? le ?_t ?'7 _lI 3ó 4.t 42 43 44 Til! î?l pLr: s';tijrìriJlr-j IF Ttil t-ì:s Is D'0= tG\ 577C ì1= cIùO Ê.2= ,St0 R3= óC 4,IJ I] TH: T,iìTÞLi S f !ìUCTURI: 1F 'TH: CIJAL D=STGN IS ttí)= 9?2C ì l.= 8'1i)0 Ã2= Btq R3= óO s:j JTH CLl';s= t7 8.8 8Lfì.K ¡:S IG\l Dr.,!: Tl î. C. BCSF i,,JïÎH pÀ?t1l:Ti:çS V=/+5 | K=12' LAvBCÁ=

1: 1ó l1 2l 25 26 27 31 35 36 37 4t 45 L3 )_? 7) 25 28 32 11 76 39 4? 43 3: 1ó t9 20 ?4 ?6 ?a 30 ?4 36 39 40 44 4: .13 t4 l5 17 ?O ?3 28 31 74 39 42 45 52 t3 11 I5 18 2t 24 29 32 15 37 40 43 llr I't 19 72 25 ?.1 iC 33 38 4L 44 7 2 l0 Ìl t? L1 13 11 30 3t 32 43 44 45 ?: li) lt t? ?o 21. ?2 3i 34 35 37 38 3S II t2 23 24 25 27 2B 29 40 4t 4? ç 17 20 2j 21 30 13 )1 .40 4V tl: 1 .I a t3 2t 24 28 31 34 38 41 44 2 I l-c ?2 25 21 32 35 39 42 45 13: + 6 t7 ls tq 27 21 ?c 37 38 t9 l!+: + î 20 ?l 22 31 3l 37 40 4t 4?_ l5: 5 ti 73 2+ ?5 3V 34 75 41 44 45 ) 't 17 13 Iq 2C 2L 22 Z3 24 25 4 1 lc t3 1ó 2l 25 ?8 t0 3-q +7 19: 7 l1 13 L6' 7_2 23 2.a 31 V7 L4 .t1 i ?¡ f't 1ê J. I tt lC 14 tó 19 24 31 V3 31 42 2 I 11 14 t6 t7 ?_5 3?_. 1+ 33 40 't I L2 14 1ó t8 2.1 30 35 39 4I c l0 15 1ó t9 ?2 27 i+ 40 45 24" t tl 15 ló le zo 23 35 4t 43 5 9 L? tt t6 17 2l 2S 31 42 !+4 2 3 ?7 23 29 30 3t 32 33 34 35 27 z t ç r0 13 le 23 26 3t 35 38 40 + I 11 13 r7 24 26 32 33 39 4L ?.9| 1 5 t? r3 18 25 26 30 t+ 37 42 3C: 3 6 7 l0 rq 17 ?_2 26 29 3+ 4L 43 3[: I å 1 tl L4 18 20 26 21 35 42 44 12t .2 5 7 t2 14 tç 2t 26 28 i3 4C 45 l3: z b I l0 15 20 25 25 2e 32 37 44 i4: a 4 I l1 t5 2l ?i 26 29 30 38 45 35: j t2 L5 22 2q 26 27 3L 39 43 36 2 2 3 37 38 39 40 4t tr7 ¿'1 44 45 31 2 5 Ê. 10 t3 l8 20 ?.9 31 36 4t 45 31r o a lr t3 19 21 21 34 36 42 43 lê. 4 e t7 13 t7 ?2 28 35 36 40 44 9 l0 t1 ?.t 23 71 32 36 3? 44 4l: I a I ll 14 22 ?4 28 30 35 3? 45 a?: ? + c L2 t4 2C 25 2a 3t ió 33 4i +3: ?. 5 7 10 15 t7 2+ 30 15 76 38 42 +4'" i '] 1 ll 15 lq 25 rl 17 36 39 40 45: 1 { t? t5 1.-l 71 17_ 34 V6 ?1 4l THï I'i IpL:l STRt]CTUql ílF rHE DiSIGN IS ?í)= 5CÁ,1 {t: çó*0 R2= 21O R3= 24i) C:SIG I I5 (!111\GLY} SELF-í]IJ^L SlI I ÏIJ \S S= t5 'L 8.9 3LI]CK . ES¡G¡I DI'I: TiI R,. A. KII\GSLEY AND R. G. STANTSN hr:TH DAR{'1ä1ÊçS V=45r K=l2r Lâ¡,tBDA= 3

t: ? 3 4 5 6 t2 1? 22 27 31 i2 33 3 4 5 -t tl ls 23 28 34 35 36 l: 2 4 5 I 13 16 24 29 37 38 39 2 3 5 ? t4 t9 2t 30 40 4t 42 2 3 4 tO 15 20 ?,5 26 43 44 45 1 e I lo ll ló 2l ?6 31 12 33 6 I c l0 L2 l3 23 23 37 40 43 e 7 9 10 13 17 2t 29 34 41 44 9: 4 . 1 I 10 t4 lc 22 30 15 38 45 l0: 5 t 7 B 9 15 ?_O 7_5 21 35 39 42 11: ? t> rz 13 14 15 16 2l ?6 34 3' 36 12: I 7 Ll li t4 15 Ll 2? 21 37 40 43 13: l q tl L2 14 15 l9 24 2ç lt 42 45 !+ q 1+: 1l I? Li t5 tc 7_3 30 3? 3e 44 15: 5 I 0 11 t2 t3 14 zo 25 28 33 33 4l 1á: I 6 Ii l7 13 1(} 2C 2l 26 t7 38 39 11: I 1 12 1.; l8 tq 20 ?.? 1t ,+ \L ++ 18: ? 1 Ii lb L1 to 20 ?7 28 31 4? 45 l9: ¿' .) !1n 14 l-r !q )^ ?L ?n ?? tÁ !,a ?12i1 c t5 1ó 17 l3 l? 25 29 32 35 40 2l z 4 ó 11 1ó 22 27 ?4 25 26 40 +1 42 ?22 I ) L2 l7 2t 23 24 25 27 15 33 45 23r Z 7 14 18 21 22 24 25 ?8 32 3S 44 ?.4. 3 I ll t9 Zt ZZ 23 25 29 33 3ó 43 25': 5 lC 15 2C 2L 22 23 24 30 31 34 37 ?-62 5 ô l1 1É 21 27 28 29 30 43 44 45 ?12 1 lù 12 17 22 26 ZB ?.9 3C 36 39 t+2 21r 2 7 15 l8 23 26 27 29 3C 33 38 4l 2,1. 3 3 Il 70 24 25 21 2g 30 ?2 35 40 30: 4 3 L4 t9 25 25 21 2S 29 3l 3q 31 ll: I /, 13 13 25 30 32 37 34 31 42 45 ?2: | 6 l+ 20 23 29 3t 33 35 t9 40 44 33: I 5 t5 19 24 28 31 32 36 l8 4L 4a 142 2 I ll 17 25 30 3t 35 36 37 4t 44 ?5: 2 ? ll 2C 22 29 32 34 36 38 40 45 ló: Z LC ll 19 24 27 33 34 35 39 42 43 372 1 1 t2_ t5 25 30 31 74 38 39 40 43 33: 3 q 15 1ó 22 29 3j v5 37 39 4L 45 3?t 1 lc 14 16 23 27 22 35 a7 38 42 44 40: t+ 1 l? 20 21 29 32 35 v7 4L 42 43 4l: 4 3 15 L7 21 28 33 34 38 40 42 44 12. 4 LO 13 18 2L 21 31 36 39 40 qL 45 +32 5 7 t2 l9 24 26 33 36 31 40 44 +5 442 5 : tå L7 23 ?6 22 14 3ç 41 43 t+5 +5: 5 1 ll t8 22 26 ìI 35 38 ttz 43 44 TllE TIIDLi SIFiUCTUS.E OF Tr-tE 0:StGN IS R0= îJ¿+0 Þ1= 8640 P2= 21O R3= 240 Ni S: C]'J IS (STIìCNlLY} SELF-I-'U,IL 5r'1 ITH CLASS= t5 B.l0 3L'ICK I:SIT;I.I I]t,,E IC S. S. SHRIKHAÀ¡DE ANt) N. K. S I ¡'IGH l.J IrlJ pAì¡,'1:TçeS V=45, tt,=L2t LA!aBCÂ= 3

:.t:) 6 7 8.9 l0 11 L2 13 14t t{ 15 16 t1 18 t9 20 ?l 245 22 ?3 24 25 26 27 28 29 23i 3C ,at 32 t3 34 35 t6 37 t-t i 2?4 't8q 38 39 40 4t 4? 43 44 45 62 t4 18 2i 29 3l 37 i9 45 ';na 15 t9 22 28 33 35 4T 43 .) 79 1ó ?O 24 2't 30 36 40 42 á 78 Ll Zt 25 26 32 34 38 44 10: TI ,t7. t4 I8 22 23 30 36 3A 44 lt: l0 L2 l 15 19 ?7 ?9 32 34 40 42 1.J II j 16 20 25 26 3l 31 4t 43 1l II 2 L7 ?L 21 27 33 35 39 t¡5 14: 2 tl I5 t6 tl ?2 2c l0 17 ?3 t+5 15: 2 7 11 L¿t l5 t7 27 27 33 3+ 41 42 16: ? q T2 L4 15 I7 ?4 25 3I t5 40 43 â 13 t4 15 1ó 25 ?7 3? 35 39 44 l9: ? t0 lq 20 21 2) 28 31 36 39 +4 i9: ? '!. I 19 70 1 2 a t2 l3 l:l ?t 25 27 30 31 41 42 ? T3 II] tç. 20 24 26 71 34 3A 45 7 lt r+ t9 71 24 25 30 35 33 43 23 z 5 II 15 13 22 2/+ 25 3l 3+ 30 42 242 4 t..l lá ?L 72 23 25 33 36 40 45 7 L2 t7 20 ?2 23 24 32 31 4L 44 ?-62 3 1 t2 L5 2L 27 28 ?9 3t 3+ 38 43 3 13 17 2C 26 ?8 29 30 35 t9 42 ? TC 15 13 25 21 29 31 36 4L 44 ?-9t 3 6 tl T4 ic 25 21 28 32 31 40 45 30: 4 3 l0 t4 20 22 ?1 3t 32 31 33 42 3l: 4 6 L2 Ló 18 23 26 30 32 33 39 43 3?| + q ll 17 lc ?5 29 30 31 33 40 44 7 13 ,l 5 ?t 24 2\ 30 3l 32 4t 45 3+r /: l l i5 2L 2J 26 35 36 37 3a +2 35r 4 1 i 3 t7 19 22 27 34 36 37 39 4V ló: /+ I t0 16 td 24 29 74 35 31 40 44 37 " 4 4 L2 t4 ?J 25 29 34 35 36 41 45 38: 5 l0 lt 2I 22 26 30 34 39 40 4t ìÕ: 5 13 L7 .I9 23 21 31 35 38 40 4l ll r(. lî ?1 29 37 36 38 39 4l L? 15 2C ?5 23 33 37 38 39 40 42. 5 ll t5 21 2i 27 30 34 43 44 +5 i3: 5 12 tó 1? ?Z ?.6 31 35 +2 4+ 45 +r'2 5 i0 t7 19 ?5 29 12 36 42 43 45 +5: 5 í, tl I.t 2L ?+ 2e 33 37 42 43 44 Ti'II .IÎIÞL: STAiJCTUR: OF TIlI D!S IGN IS R,)= 5i-r4ll c l= 3640 p..2=, 27O R3= 240 D:SIC^: IS (iTIÍ\GLY} S:LF-If'AL s1ì I r r-.1 cL dss= l5 8.11 qL,'rt( D5î I;,J I,Jf 1.,¡ ,rrllr Kr\tGsLsy AND R. G. srA,\rofl "iij;l;r;i=v,=,;u*.,.n...r K=II, L¡ÞjBf A= 2 l: I 44 /+i 17 .4q . ¿.9 50 5l 53 2) 3+ 3!, atr 4t 4? 54 56 71 _?.t 1? 3i 43 45 4^ ;; :-":, :î ),) 38 17 44 47 12 33 37 38 ,? ;; ?.1 39 40 46 50 ?5 26 2E 39 ;6 42 43 5t 55 54 -?. : 2l ?-.1 2+ 27 3q ?) 4t 45 5+ 55 ,t- ?-2 ?.3 25 4C 4¿19 1 I 45 +7 49 52 ?-5 77 28 33 34 't: a 1 15 50 52 6; l9 ¿1 28 34 I?: .) 46 17 lil 2 3 )-4 ?-5 Zz 11 ,7 55 ll: II c.r t7 43 52 ì I 22 ?i 31 16 5t 12 J 40 ¿,3 q3 55 2r. ?_Z 26 30 :? lì: II 7 35 37 11 4a qô .. 4-t ¿5 2:) 30 l4: t4 r_ì ?0 23 36 39 44 ;; t 5: l5 ?2 zc 31 33 +l 4'+ ;; . 1 2-) 2t 27 31 32 i5: is l? 2ü 1+ 39 /+g 1; -l 25 30 33 18 42 5? 1! 1{, L1 40 45 ) I 4t 4i 50 5r ;; 1.? 14 t3 39 À 39 \2 4g l? ll la. lç 3e 52 ;4 )'\. 4L 44 +6 49 s5 1q t5 15 2C 35 36 77 qq i L.a l- ?1 22 5L *) 7 75 +4 46 5i ;; tl L2ß22 12 31 ) 7 34 45 5I ;; IC l+ ?3 3C 34 L? 5 3 i0 39 51 5? ') 'ta . tl 24 3I 5 1 q 31 35 42 4s ;; -. tl 25 3t ), t¡ 37 39 4L 48 ;; 12 13 26 29 31 :_¡ a 34 40 47 ;; ,a. ll tq 2'l 3) 37 5 4 ? t+ 42 47 ;; 23 30 32 36 +o 45 ;;

8.12 ) + l3 l4 7t, 21 29 49 52 5t 55 3C: ì t2 l3 1ó 23 27 23 30 46 48 56 3 1_1 11 24 25 ?.6 vt 45 46 50 i2-. 3 4 Lc 15 22 28 32 AI 42 53 54 * 3 L4 1ó 21 22 24 ?3 43 47 48 'r 't ... 7- 3 9 l:1 )', ) < 34 38 44 56 2 _l t2 2a 2+ 35 39 40 51 53 ?12 ? tc l1 ll 20 2t 28 36.38 47 50 37: 1l lf, L2 20 25 21 37 q3 44 +5 38: 2 t l5 13 2' r+ 3ó 38 4? 51 ?.o. 1l l3 1S t9 27 35 39 47 : 45 ') 7 L7 26 21 35 40 41 48 /+1: l4 17 19 ?5 32 4t 47 56 5 7 L6 13 2t¡ 27 32 42 44 50 a 5 II 17 23 3V 11 43 4e 14 /,42 l 1.3 14 t9 2T 3q 31 40 42 44 2- é t6 22 23 3l v7 !9 45 52 4r i { 9 rr ?t 30 31 46 51 54 iii ? ; ç 26 27 33 36 39 4t 41 l0 15 l8 25 3C 37 40 48 55 +q: 1 t2 1? 24 29 27 38 4Z 49 'i L!- JO +¿ iU )> ., 1: ) 1ù Lt* 17 2? 21 35 38 46 51 1 l7 lt 2L 2.q 45 52 56 ¡ tt t6 2t ?_1 25 29 32 35 53 54'.)'+. I l I I) r8 2-O 25 32 43 45 54 |:::. q a t lç ?o 22, 2c. 48 50 55 5t¡| t 4 5 1l ?o ?4 30 34 41 52 5ó Trl:: T?, I )L: Sl'!UCir-la.t ûF Tll! Dir S IG¡l IS Rl=1318C ll= 9240 R2= C I'): S I GII IS (iTPÎii;LY} S:LF-I]tJAL STl I T!-I C L,I.3 S = ?o

8.13 BIBLIOGRAPHY BIBLIOGRAPIIY

t 1l H. I'. Baker, PrincÍpl-es of Geometry I, Fóundations, . Cambridge, 1922. L 21 It. F. Baker, Principles of Geometry II, plane Geometry, Carnbridge, 1922.

t 3l II. tr'. Baker, Principles of Geonerry III, Soi_id Geometry, Carûbrídge, 1923. l4) H. F. Baker, forrlåt j-ons,

t 5I W. C, Blackr¡elder, On Constructing Balanced Incomplête Block Desígns from Association Matrices with Special tion Sc T\,ro and Three Classes, J. Conb. Th. 7 (1969), 15 - 36. t 6I W. C. Blackwelder, ConstructÍon of Block Desisns from Association Sc Proceedj-ngs of the Second Chapel Hill Conference on Conbinatorial Matheaatics and í¿s Applications, 1970, 28 - 36, 17I R. C. Bose, Mathenatical- Theory of the Symmetrical Factorial- Design, Sankhya I (1947), LO7 - 166.

18I R. C. Bose and T, Shimanoto, Cl,assifícation and Analysis of PBIBDTs with two associaEe clâsses, J. Amer. Stat. Assoc. 47 (1952), 151 - 184 t9I R. C. Bose, I"I. H. Clat!¡orthy, and S. S. Shrikhande, Tables of P"rti^lly Balanced Designs wffi

tfol R. C. Bose and I. M. Chakravarti, Hernitlan Varieties 1n a Finite Projectíve Space ?c(4,izf, Cæ. .¡l ll"ttr-Tg

[ 11] P. Denbowski, I'inite Geometries, SprÍnger, New York, 1968.

l12l L. E. Dickson, LÍûear croups, Dover, Liepzig, 1901. [13] ï,. E. Dickson, c. A. ltiller, and II. I. Blichfeldt, Finite Groups, Ðover, Chicago, 19L6. l14l A. L. Dulnage and N. S. Mendelsohn, A systenatic Method for Combinatorial Counts, Studíes in Applied MathemaËics A ÕtÐ;105 --TiI. - i15l I^I. !..Edg:: Somè Inpl-icárions of rhe Geomerry of rhe 2l-Poinr p1âne, Math. Zuir.@

116l Sir R, A, tr'isher and F. !I. yares, SÈarisrical TabLes !o_r Biglogi.cal, Agriculrural, and. MãAñãï-@f 0*r ea SO:. [17] M. Hal-l Jr., The Theory of Groups, MacMillan, New York, 19591-

[f8] M. HaLl Jr., Combinarorial_ Theory, Blaisde]_l, Irraltharn' 1967'

t19l IC. Hal-l Jr., R. Lane, and Ð. Ìlales, Designs Derived f ¡oü lgrnuraríon croups, J. Conb . Th--( ltTO) ,A] 12 - 22.

[20] A. llenderson, The Ti/enty-Seven Línes upon the CubÍc Surface, Carnbridge, 1911.

l2I7 D. M. Johnson, A. L. Ðuinage and N. S. Mend.elsohn, and orthoeonal- Latin Can. J. Marh. 13 (1961

l22l [ü. Jonsson, The (56,I sl. and ttrales, J. Conb. Th.

L23J R. A. Kingsley and R. G. Stanron, 0n a Block Design g!,_11"Ð-, ?roceedings of the rwånillïiÌ-th Sunmer Meeting of the Canadian Mathenatical Conference, Thunder Bay, 1971, 23L - 244. 124) R. A, Kíngsley and R. c. Stanron, The Double-Six Btock pesisn (36, 15, 6), proceediiliìFiãã- ãitoba Conference on Nr¡nericaL Mathematics, I,IÍnnipeg, 1971, 581 - 588.

l25J R. A. Kingsley and R. c. Sranton, On a Cubic Surface Block Design (45, 12, 3), ibid., 45L -756. 126l R. A. Kingsl-ey and R. G. Stâ.nÈon, A Survey of Certain P.alaqced Inconplete Block Ðesígns, p.oce.dfn# th" third Southeastern Conference on Combínatorics, "fGraph Theory and Compurirrg, Boca Raton, 1972, 305 - 310. l27l N. S, Mendelsohn, Inrqrsection s ín Studies in Pure MathemaËics, Ac c Press, J.9 71, tr45:150.

[28] l.{. Nerrrman, Invariant Factors of Conbinatorial Mat¡ices- Israel J. Ma t29 . B. Quist, So¡ûe Redarks Concerning Curves of the Second Ðegree in a Finíte Pl-ane, Ann. Sci. Fenn., ser. A, I. Math.-Phys. no" 134, Helsinki, 1952.

t30l Il. J. Ryser, Conblnatorial Mathenatics, Math. Assoc. Amer., Rahway, 1963.

t3fl L. Schlafli, An AttemÞt to ÐeËenmj-ne the TÍrenty-Seven l,ines upon a Surface of the Third Ordèr, etc., Quarterly J, Math, 2 (1958), 55 - 65 and 110 - 120.

1321 B. Segre, Ovals in a Finire Plane, Can. J. MaËh. 7 (19s5), 414 - 416.

t33l B. Segre, Lectures Ln Modern Geometry, Rone, 1961. i34l S. S. Shrikhande and N. K. Sineh, ConsÈtuctins Svr¡met ock ,4,25-32. t35l E. Spence, Sone Netr SyÍmetric BIBD's, J. Conb. Th, r1 (197r), 299 - 302. t36l R. G. Stanton and D, A. Sprott, Block Intersections 1n Salanced Inconplete Block DesfEãËllããl- MãEE- B-i 1. (re64) ,

1371 R. G. Stanton, Abstract Al-gebra and its Applications, I{ater1oo, 1962. t38l R. G. Stanton, Sone Types of Statistieal Designs, Canadian Mathematícal Congress Seminar, Saskatoon, 1963. t39l J. J. Sylvester, Note sur 1es vingt-sept droítes drune surface du troisiéme degre, Compres Rendus 52 (1861)- 977 - 980 [40] K. Takeuchí, A Table of D.Llference Sers ceneraËing B"l".c.d Irr.o t41l K. Takeuchi, On Èhe Const Desisns, JUSE I0 (f963), 48. l42J B. L. Van der !üaerden, Geonetrie II, Díe geraden Linien auf den llyperflachen des p_, i43l J. l[alJ-is, Tr,¡o New Block Ðesigns, J. Comb. Th. 7 (L969), 369 - 370. t44l l[. D. Wal,]-is, Some Notes on Integral Equivalence of Comblnatorial Matrices, Israel J. Marh. l0.(1971), 457 - 464. t45l I. S. I[oods; Higher Geouetry, Dover, New York, 1922.