This dissertation has been 62-2142 microfilmed exactly as received
JOHNSEN, Eugene Carlyle, 1932- MATRIX RATIONAL COMPLETIONS SATISFYING GENERALIZED INCIDENCE EQUATIONS, AND INTEGRAL SOLUTIONS TO THE INCIDENCE EQUATION FOR FINITE PROJECTIVE PLANE CASES OF ORDERS n = 2 (mod 4).
University Microfilms, Inc., Ann Arbor, Michigan
JOHNSEN, Eugene Carlyle, 1932- 62-2142
The Ohio State University, Ph.D., 1961 Mathematics
University Microfilms, Inc., Ann Arbor, Michigan MATRIX RATIONAL COMPLETIONS SATISFYING GENERALIZED INCIDEUCI
EE EAT IONS, A'JD INTEGRAL SOLUTIONS TO TUE INCIDENCE
EQUATION FOR FINITE PROJECTIVE PLANE C.^SES OF
ORDERS n = 2 (ircd li)
DISSERTATION
anted in Partial Fulf i 1 lr.ent of tr e Reqoirer.ents for the Txa Sector of Fr.iloscrhy in the Graduate School of The Ohio State University
Dy
Evonne Carlyle Johnsen, P. Chen.
The Chio State Ur,iveraity 1961
Approved by
(J'
Adv i ser ;epartrr.dit ofi-«r< thematic-. ACKNOWLEDGMENTS
I wish to express my gratitude to Professor H, J. Ryser of the Department of Mathematics at the Ohio State University for his motivation of, and stimulation of■insight intofthe problems treated in this dissertation and for his valuable suggestions concerning this presentation. TABLE OF CONTENTS
page
I. Introduction...... 1
II. Matrix Rational Completions Satisfying Generalized Incidence Equations
1. Introduction ...... 5
2. Basic Theorems ...... 6
3. The Class of Matrices JB ...... 4 10
k« Rational Completions ...... 15
III. Integral Solutions to the Incidence Equation for Finite Projective Plane Cases of Orders n = 2(mod k)
1. Introduction...... 23
2. Skew-Hadamard Matrices and Designs ...... 26
3« Construction Theorems ...... 3k
IV. Appendix - Some Results on the Integral Solutions to the Incidence Equation for gcd(k,k-\)» 1 and k-X even 55
V. References 67 X
I. Introduction
Let us consider the following problem. Let there be v ele ments x . ..., x and v sets S., .... S such that every set 1 v 1 v contains exactly k distinct elements and every pair of sets has
exactly X distinct elements in common. To avoid degenerate sit
uations we stipulate that 0<\ v, k, X confiauration or design. For what values of v, k, X are these ccnfiaurations possible? A related question asks for the num ber of nonisomorphic configurations for a given v, k, X; however, this question poses a somewrat more difficult problem which is not of direct concern to us here. vJe can give an equivalent character isation of a ccnfiduration in terms cf a matrix A ** [a. called » r its incidence matrix, by writing the elements x^, • ••> xy in a row and the sets S f ..., Sv in a column and setting a « 1 if ^ J x. is in S. and a, , ■ 0 if x, is not. This matrix A. of J 1 ij j order v, is composed entirely of 0's and l*s and by the con ditions of the problem is easily seen to satisfy the matrix equation (1.1) AAT - (k - X) I + X S = B , where is the transpose of A, J is the identity matrix of order v, and S ■ [s..], s.. * 1, 1 < i, j f v, Equation (1.1) J J is known as the inc idencc equal ion for a v, k, X design, Ryser showed in [9] that X (v - 1) ■ k (k - 1) and that A is norma1, T T i.e., AA * A A ■ B. Two special classes of v, k, X designs are of particular in- 2 terest. The first, where X ■ 1, k « n + 1, v * n + n + 1, a > 2, is a configuration which is better known as a finite pro.jective plane of order n ■ k - 1. So far these configurations are known to exist when and only when n is a prime power, and known not to exist for an infinite class of other values (see Theorem 1.1 below). The second, where X ■ m - 1, k ■ 2m - 1, v » Um - 1, i s a configuration known as a Hadamard design, which characterizes a Hadamard matrix of order Um. These are known to exist for various infinite classes of order Um - 1, and the conjecture is made that they exist for all values of order Um - 1* Configurations are known to exist for many other values of v, k, X as well. Considering A as a matrix with rational entries, equation T T (l.l) with AA m ALA asserts that B is rationally congruent to the identity. Here we apply the Hasse-Minkowski theory for the rational congruence of two integral symmetric matrices. Using the fundamental Hasse-Minkowski theorem, Bruck and Ryser [3] proved the following’nonexistence theorem for finite projective planes. Theorem 1.1. If a projective plane of order n = 1, 2 (mod U) exists then the squarefree factor of n cannot contain a prime factor p ; 3 (mod U). Under the assumption n = 1, 2 (mod U) the condition that the squarefree factor of n does not contain a prime factor p ~ 3 (mod U) is equivalent to saying that n « + b^, where a, b arc integers. Later, Chowla and Ryser [U], using elementary methods and not the Hasse-Minkowski theory, proved the following nonexistence theorem for general v, k, X configurations. Theorem 1,2. IT a v, k, X configuration exists then (a) jjf v fs even k - X must be a square, (b) JT v _is odd the equation o 9 ^ 2 x ■ (k - X)y + (-1) X z must have a nonzero integral solution. So far, this theorem accounts for all known excluded configurations. For finite projective planes the theorem reduces to Theorem 1.1. Specializing the theorem to Hadamard designs, however, yields no nonexistence theorem since with v odd and for any integral x ■ y ■ z / 0 the equation resulting from (b) is always satisfied* If wenow consider A as a matrix with integral entries, then equation (l.l) with AA^ ■ AIA^ asserts that B is integrally re presented by the identity. Unfortunately, the theory of integral representation is, at present, insufficiently developed to handle this situation. If it were, we would probably still, as in the situation of rational congruence, get for v, k, X designs in general essentially a nonexistence theorem, since we are still in a situation more general than the 0,1 situation. However, for the special class of v, k, X designs where k - X is odd and gcd(k, k-X) is squarefree, Ryser has shown by theorem 2.2 in [10] that any complete set of criteria for integral representation and nonrepresentation by the identity would effectively decide the existence as well as nonexistence of a design. The foregoing discussion gives the setting for what is to follow. In parts II and III we will go deeper into the situations in the preceding two paragraphs, in that order. In part II we are motivated by two questions which were previously asked and answered. When is there a normal rational solution to the incidence equation? Given a design consistent start of r complete rows or columns of O's and l's, when is this start rationally completable to a normal solution to the incidence equation? The first question was answered for a special case of finite projective planes by Albert [1], and then later both questions were answered completely by Hall and T Ryser [7]. These results together with the question - If AA « B, T ~ what is A A? - gave insight to and motivated an extension of these results both to a class of matrices of which B is a member and to a more general situation where we have a consistent start of r rows and s columns. In part III we continue the investigation of Ryser [10] into integral solutions to the incidence equation. There we limit the investigation to nondesign integral solutions for the finite pro jective plane case of orders n = 2 {mod h) and obtain construction theorems for certain values of n. A special class of Hadamard de signs called skew-Hadamard, on which these constructions depend, are investigated there. A discussion of some fragmentary results pertaining to a somewhat more general class of v, k, X designs is relegated to the Appendix. II, Matrix Rational Completions Satisfying Generalized Incidence Equations 1, Introduction, Albert [1] proved for the finite projective 2 2 plane case of orders n • a ♦ b , where a, b are integers, that there exists a normal rational matrix A satisfying the incidence equation (l.l). The proof was constructive and from it A could be obtained. Later, Hall and Ryser [7] extended this result to all design cases whose matrices B are rationally congruent to the identity. In that same paper they also proved a stronger theorem, which stated that if the matrix B of the incidence equation for any v, k, X is rationally congruent to the identity and we have a 0, 1 entry start in r complete rows (columns), 0 < r < v, whose row (column) inner products are consistent with those of a design, then that start can be rationally completed to a matrix A of v rows (columns) which is a normal solution to the incidence equation. These proofs made some use of the combinatorial aspects of the problem, but for the most part they were obtained by using matrix theory. Meanwhile, Ryser i.11']» in investigating the integral solu tions to (1.1), toint'i- the way to a generalization of U at equation in the proof of his thtorvr h.h. This theorem will be discussed later in connection with hh; problem of the integral solutions to (1.1), but its proof showed that if a rational matrix A satisfies T (1.1) then A satisfies a generalization of (1.1) where B is replaced by a matrix in a class JB of matrices over the rationals of which B is also a member. In this part we generalize some of the preliminary results in [7] and then go on to generalize the above mentioned main results in that paper. Finally we top this off with a result whose conjecture is readily motivated by what had previously been done, namely, that if any matrix in the above mentioned classJB, for a given v, k, X, is rationally congruent to the identity and we have a rational entry start in r rows and s columns, C < r, s < v, whose row inner products and column inner products are consistent with the entries in two matrices and B0 in the class JB respectively, then that start can be rationally completed to a matrix A of order v T T such that AA - and A ‘A • When the rational entries in t hie start are 0's and l's and B, » B *» B, then we net n rev 1 2 rational completion result related to v, k, X designs. In what follows, capital letters will generally denote mat rices, We denote rational congruence b y * and rational incongru ence by . For the most part we will be specifically concerned Basic Theorems. We first state a theorem which was proved in [7]. An equivalent restatement is given parenthetically. T* Theorem ?.l. Suppose AA1 - $ D^. Here the matrix A orlier n and non singular. The matrix j_s of order r and B, is of order s, where r + s » n. Let X he an arbitrary T T " >: n (nxr) matrix such that XX • {X X « D ^ . Then there exists an n x n matrix Z having X as its first r rows (r col umns) such that ZZ^ - Dj • (Z^Z ■ Dj • D^). This result holds for all fields F of characteristic 4 2. m Using this theorem we can prove the following theorem. Theorem 2*2. Let X and Y be two m x n matrices with en- tries in a field F of characteristic i 2 such that XX « YY - Z. If rank X ■ rank Y ■ rank Z, then there exi st,s an n x n ortho gonal matrix Q with entries in F such that X Q - Y. Proof: Let the common rank of X, Y, and Z be r > 0. If r - 0 then X * Y ■* 0 and for Q « I we obtain X Q ■ Y, Now suppose r > 1. Since Z is symmetric there exists an r x r principal submatrix Z^ in Z of order and rank r < min (m,n). Then we have T T X X Y Y r r r r where Xf arid Y^ are corresponding r x n submatrices of X and Y respectively. We have r ■ rank Z„ < rank X . rank Y < r r — r r — w lie nee rank X rank Y Now let P be an m x m permutation matrix which move: <\ nun to the first r rows of X arm respect:vcxy. 'iris yield: 1 ^ r X “ anti P Y « X, Y Let X be an (n - r) x n matrix whose row vectors are a basis for r the orthogonal complement of the space spanned by the row vectors of X . Consider r Vie have X X o o If r - n then X is nonsincular. Now let x,, ..., x be the r 1 r row vectors of X and x,, .... x„ _ those of X , where r < n, r 1 ii-r p * Suppose there exists a relation (2.1) + arxr+ h1^1+ + bn-r*n-r ’ °> a p bj E F ‘ Now taV.ing the row inner products of (2.1) with x., 1 < i < r, we obtain {e. 2) a^(xjjX^) . + ar ^ i > xr) “ °» 1 < * < r • Now (2.2) is a honooeneous system of linear equations with matri: ' (x ,x.) ] - X X Z , where Z is rionsinqular. Hence (2.2 ) 1 J r r r r has only the trivial solution a^ 0. Then (2.1) be- comes ft x. + ... + b x “I "1 n-r n-r whence b - ... - bm ■ 0 since x , ... x are linearly in- 1 n-r n-r J dependent. Hence x,, ..., x . x - .... x are linearly inde- 1 ■ r* 1* ' n-r J pendent which means that again X is nonsingular, Now T ° Y ■ 2 ! hence by Theorem 2.1 there exists an n x n matrix r r r * Y with entries in F having Y as its first r rows such that o r T T YhY ■ XX A • Then the n x n matrix 0 0.0 0 ' Q - X** Y o o is orthogonal, and since X0Q ■ Yq, Q • Yp. Let x^, x^ be the row vectors of P X and y., • *», y those of P Y. Now J- m for any l Let r X 2 T u.x. , y - 2 1 v y. , u., v e F . r+t i.l 1 1 r+t j.i J J 1 J \ - k Then taking row inner products xk) anf* ^r+t* ‘V. 1 < k < r, we obtain (W * V > ■ ui(5i’ *k> ' " d r r - 2 1 v v v ' - 2 T Vj(x ,x ) . j-i j j * j-i J j ^ T ! T since X.< * 'A' J hence r ^ ' (,Ji" v j) (^i» xk) ■ r'» 1 < k 5 r . i»l daw (, ,3) is a homogeneous system of linear equations with matrix [(x., x, )] » Z , where Z is nonringularl hence (2.3) has only i v. r r the trivial solution u - v. ■ Of 1 < i < r. hence u. » v^, 1 < i < r, and for any 1 < t < m - r 10 r , r- *” t<5 ' ui *l 8 ' i Z Vi ‘ ’ whence PXQ • PY or X Q » Y . Corollary 2.3* Let X and Y be two m x n matrices with m entries in a formally real field F such that XX • YY • Then there exists an n x n orthogonal matrix Q with entries in F such that XQ - Y» Proof: Let x,, x be the row vectors of X with 1 1 m components in a formally real field F. Now the rank of the Granian [(x., x.)] of these vectors is equal to the dimension of 1 J the space generated by the vectors (See [£], pp 2L6-7 for a dis cussion of Gramians)* Fence for a formally real field F, if T T XX ■ YY » Z , then rank X - rank Y » rank Z and we have the corollary by Theorem 2.2« 3* The class of matrices |B« tfe define a v x v matrix B(u), u » (u1# as B(u) C X u u + (k - \)I - [X u ^uj + (k - X)6j j ] , where the u^'s are rational and satisfy v 2 ^ _»T Z u. - u u - v , i-1 S. . is the Kronecker delta, and where (v - l)\ ■ k(k - 1), ^ J 0 B(-u) - B(u), and that B(x) » B(u) implies x ■ * u. Let us denote the class of all such matrices B(u) for a fixed set of v, k, X parameter values by JB. If A is a v, k, X design T matrix then AA » X S + (k - X)I Z B, where S » [s j j], s.j - 1, l j ^ - and Y Y » B (w) for v x v rational matrices X and Y are called generalized incidence equations. Lemma 3,1. Any two matrices in JB are rationally orthogon ally congruent. Proof: Let B(u), B(w) be arbitrary in JB. Now jk T ^ j j u u - w w ■ v. Hence by Corollary 2.3 there exists a v x v rational orthogonal matrix Q such that u Q » w . Then ip Q B(u)Q ■ Q^(Xu u + (k-X)I)Q » X Q*^ u^ u Q' + (k-X)Q^Q T T - X(u Q) (u Q) + (k-X)I - X w w + (k-X)I - B (w). Thus B(u) — B(w) orthogonally. By Lemma 3.1 either all or none in JB are rationally congruent to the identity I. Hence we may write without ambiguity either JB s I or JB ^ I. Lemma 3*2. Every B(u) in JB is nonsingular; 12 Proofr Wc have B(u)(k-X) [I-Xk u u] • [Xu u+(k-X)I](k-X) [I-Xk u u] “1 tT». 2 2 >t .t —2 tT t • (k-X) [Xu u-X k u uu u+(k-X)I-Xk (k-X)u u] 1 2 — 2 2 • (k-X) [Xu u -X k v u u-Xk (k-X)u u] +1 2 ^ and replacing k- X by k - X v we see that B(u) is non- 1 1 g T singular with B (u) - (k-X)" [I-X k u t], Since by Lemma 3.1 any two matrices in JB are rationally orthogonally congruent, they also have the same determinant, whence all the matrices in JB have the same determinant. By computing the determinant of B ■ X S + (k-X)I we easily find that this value is k2 (k-X)V t Theorem 3»3* Let X be a v x v rational matrix j ^ ip ^ satisfying XX • B(u)* Then X X * B(w) where w « e k u X, £ * 1> “1# j ^ Proofi We have XX ■ B(u) where B(u), and hence also X, is nonsingular. Hence we may write T -1 ^ -1 X B (u) - X , which by Lemma 3.2 becomes T o T i (2.1) X [I - X k u u] - (k - X) X" . Multiplying (2.1;) on the right by X yield* (2.5) XTX - X k"2 (u X)T (u X) - (k - X) I, and setting w » e k~* u X, e • 1, -1, in (2.5) we obtain J X X ■ X w w + (k-X)I • Now _k _,T -2-k T-kT -2-kr -kT-k , s ,-kT w w ■ k uXX u m k u[X u u +(k-X)l]u .“2 -» _»T v-k -*T • k [Xuuuu ♦ (k-X)u u J o 2 k v[k - k + k] • v • Hence X^X • B(w) and the theorem is proved. From this theorem we can derive the following corollaries. Corollary 3,tu For a v x v rational matrix X, any two of the following three conditions imply the third; (a) XXT - B(u) (b) XTX - B(w) (c) uX - e k w, e ■ 1, -1, Proof: By Theorem 3.3(a) and (b) imply (c) and (a) and (c) T imply (b). Now assume (b) and (c). Applying Theorem 3.3 to X _ k we see that there exists a z such that (2.6) XXT - B(z). Applying the theorem again to (2.6) and (b), we have (2.7) z X • k w, ■ 1, -1, and combining (2,7) and (c) we have Ill z X ■ e2 u X, e2 - e1 e. and since X is nonsingular* 2 • ^2 u» ®2 * 1» "1 • T ^ Hence XX ■ B(u), which is (a)* Corollary 3.?. For a v x v rational matrix X, any two of the following three conditionsimply the third; (a) XXT - B(u) (b) XTX - B(w) -fc T ^ (c) w X - eku, e-1, -1. T Proof; This is Corollary 3.1; with X and X interchanged and u and w interchanged. Corollary 3.6. Let X be a v x v rational matrix such that XX ■ B(u) and X X * B(w). Then u X • e k w j_f and only ^ T ^ if w X • e k u, where e ■ 1, -1. Proof; By Corollaries 3.1; and 3.5 we know that uX ■ e^k w -> T and w X ■ e k u, e,, e - 1, -1. Then 2 1 2 2-* T -» T . eiE2 U " £1^ - uXX " u ® tu) « u(X u u ♦ (k-X)I) . . -k 2 ■ (X v + k-X;u - k u Hence e, ■ 1 and e, - e - e, 1 2 1 2 Using what has been proved up to here, we can now prove the following theorem. Theorem 3.7* Suppose JB * I, and let B(u), B(w) be arbitrary in JB. Then there exists a v x v rational matrix A such that AA^ - B(u) and A A ■ B(w). Proof: Now B(u) i I, Hence there exists a v x v rational matrix C* such that CC^ ■ B(u). By Theorem 3.3* C^C ■ B(x) for _» _»T some B(x) in JB, Now x x - w w - v, hence by Corollary 2.3 there exists a v x v rational orthogonal matrix Q such that x Q - w. Let A ■ C Q. Then T T X X T AA - C Q(CQ) • C S Q C - CC - B(u), and since by Corollary 3«U uA«uCQ»ekxQ»ekw, e • 1, -1, ■ % T we have by that same corollary that A A ■ B(w), The following result of Hall and Ryser [7] then becomes a corollary to this theorem. Corollary 3*8. Suppose B = B(*f) ^ I. Then there exists a T T v x v national matrix A such that AA « A A ■ B, L. Rational Completions. Let x • (x^, ..., xv )» where the Xj's are rational, satisfy x x • v (i.e. B(x) eJB). We then define xm - (x . ..., x ) and x^ ■ (x * m i' m m v-m*l ' v 9 where i and t suqgest "initial" and "terminal" with ref erence to the subset of m entries taken from x. Bfx1) and m * B(x^) then denote the initial principal and terminal principal * m x m submatrices of B(xJ respectively. We further let x . m ■ without any attached i or t, denote any m - tuple of _*T rational entries such that xm < v and define B(x ) - X x^ x_ ♦ (k-X)I • We define a T"'r - array as the portion of m m ra m * s a v x v matrix obtained by taking the configuration consisting of _ r the first r rows and the first s columns. We let j denote the r x v row submatrix and J”' the v x s column submatrix of ^™‘r , s s We can now prove a general rational completion theorem for a given consistent rational start of s complete columns of entries. Theorem L.l. Let JB i I and let B(u), B(w) e JB. Suppose Ag, 0 < s < v f h a v x s rational matrix such that A A^** B(ws). Then u • e k w , e ■ 1, *1, necessary and sufficient con- 4 dition-■ that ■■ — —there — exist ■' —a v x v ■■■rational...... matrix — ■ A containinq* A„ s T ^ as its first s columns and satisfying AA - B(u) and ATA - B(w). Proof: Let u A ■ e k w \ e » 1, -1. Since B(u) -s- I there s s T -» exists a v x v rational matrix C such that CC « B (u). We re call by Lemma 3.2 that B ^"(u) ■ (k-X) I-Xk ^u^u]. Then con- -1 sidering C A^ we have CC_1A )T(C_1A ) - A ^ C c V ^ - A7 B_1(u)A ^ b 5 5 S . ( k - x r V \ 5-xk‘2dAs)T(^s)] • (k-X) [(k-X)I ♦ X w. - X ; iT w ‘l s s s s s where 1^ is the identity matrix of order s. Then by Theorem 2,1 there exists a vx(v-s) rational matrix R such that 17 [C~^A , R] is orthogonal. Then A* (C1 )-1 (2.8) i c ' \ . r] - 1, whence by multiplying (2.8) on the left by C and on the right by CT, [A , C R] _T_T ■ CC • B(u). R C ■Now by Theorem 3»3 and Corollary 3.L there exists a B(x) e such that [As>CR] » B(x), u [As> CR] ■ k x, e# - 1, -1. T T R C ^ -k Uki we can choose x such that ■ e hence obtaining x - w _»i _.i since 6 k x ■ uA„ - e k w . Now s S s -.i-.iT -.t -.IT Jk _ T -.T -.i -.iT -.t j.tT X X ♦ X x »xx«v«ww»ww+w w , s s v-s v-s s s v-s v-s —k 1 -.iT -*i -.iT -.t -.tT -kt -.tT where x x - w w . hence x x » w w Then s s s s v-s v-s v-s v-s by Corollary 2.3 there exists a (v-s)x(v-s) rational orthogonal matrix Q such that Q ■ s* Let A« [A^jCi'.j], Then T T T T T T T T AA - A A ♦ CRQQ R C - A A +CRRC s s s a - [A ,CR] B(u), s T T R C and u A * u[A > CRQ] - [u A , u CRQ] , -ki -kt. , r-*>i •Jkt , • [ekw.ekx Q] • e k[w . w ] s v-s s v-s - f c « c k w * T ^ Hence by Corollary 3 . h t A A ■ B(w). This shows that the condition is sufficient. The necessity of the condition follows trivially from Corollary 3»li since A ■ [As, X], AA ■ B(u), A A - B(w) imply _k ^ _ki u A » e k w implies u A ■ e k w where c » 1. -1, s s It should be clear that Theorem h,l could also have been stated in terms of a row start instead of a column start. When done so the following result of Hall and Ryser [7] becomes a corollary to the theorem. Corollary U.2. Let B = B(l) A I, Suppose A , 0 < r < v, i_s an r x v matrix of 0's and l’s such that ArAr^ • BC11), Then r there exists a v x v rational matrix A having A as its first T T r rows such that AA » A A « B. Proof: The condition for the rational completability of the .k rT -ki row start to the desired matrix is that 1 A » c k 1 where r e m 1, -1, Now by ArA r « B (11) every row of A1* has exactly r k l's and v - k 0 ’s, hence the condition is satisfied. We can also obtain the following result as a corollary to the theorem. Corollary L,3« Let £ * I and let B(u) e JB, Suppose A 1" r 2* rT ^ j a * r, - arrayt 0 < r t s < v, such that A A « ®(ur ) and T ^ ^ A g A^ ■ B(wg), Then u Ag ■ t k ws, e ■ 1, -1, _i_s a necessary and sufficient condition that there exist a v x v rational matrix containing A^ a s a ] - array and satisfying AA ■ B(u). Proof: Let u A *ekw,e-l, -1, and let % «[A , X] b " ■ s s' s' a matrix such that AA • B(u), which is guaranteed to exist by Theorem U.l. Now set 1 V * > A - , X - s z 1 V * n where W and Y have r rows, and set A • [W, Y], Let A - [W, U], Now ^rT n/-iv .r rT A A » B(u ) » A A , r whence T T W W WW + YY - [W,Y] [W,U] - WWT + uuT u1 or YY • UU . Then by Corollary 2.3 there exists a (v-s)x(v-s) rational orthogonal matrix Q such that YQ * U. Set 1 ’w YQ -w36 U l u s, xa]. Z VQ z v q J r j % r Then A contains A as a j -array and AAT - A A ♦ X Q(X0)T * A AT * XXT - A / . B(u). s s s s This shows that the condition is sufficient. The necessity of the condition follows from Theorem 3*3 and Corollary 3.1j since W u A - AAJ • B(u) 2 * implies that there exists an x such that A A ■ B(x) and ^ ^ -ki -k u A ■ e, k x where x ■ e_ w , e., e„ ■ 1, -1, which implies that 1 s 2 s 1 * uA • e«k x ■ e.fc k w » e k w * e ■ e e* * 1. -1# s I s 1 2 s s 1 2 This corollary readily suggests the following theorem. Theorem li.U* Let i I and let B(u), B(w) e JB. Suppose fiF is a -arrayj 0 < r, s < v, such that Ar Ar^ ■ B(u*) and T s A A » B(w ). Then u A ■ e k w* and w Ar^ ■ eku1. e ■ 1. -1 S S s S s --- r' ' are necessary and sufficient conditions that there exist a v x v r 4_ \T rational matrix A containing A as a ^ -array and satisfying AfiJ • B(u) and A^A - B(w). _» _»,i _» rT _»i Proof: Let u A r » e k w and w A -eku_, e-1,-1, "— s s r Also let V Ar - [W,U] and Ag - Y By Theorem h.l there exists a v x v rational matrix A containing r* /vT /-*» Ag as its first s columns and satisfying A A ■ B(u) and A A - B(w). Let [I x l where X1* » [W,X’] and A ■ A . By Corollary 3.1i u A • e k w, s s ^ e# ■ 1, -1, and since ->i ^ _» _»i e.. k w - u A - u A - e k w * s s S s and s > 0, we have ■ e . Hence by Corollaries 3.$ and 3.6, —* (2.10) XXT - UUT Hence for X * U P - _.t and R ■ -*t w w ■ V-s 4 v-s of size (r+l)x(v-s) we have from (2.9) and (2.10) that T T PP » RR . By Corollary 2.3 there exists a (v-s)x(v-s) rational orthoqonal matrix Q„ _ such that PQ „ « R. We now construct the V-s » ^ v-s ■ v x v rational orthogonal matrix Q » Ig • Qv_s> where I is the s x s identity matrix. We set A ■ A Q, Then hence by Corollary 3.U A^A » B(w). This shows that the conditions are sufficient. The necessity of the conditions follows trivially r , • X ^ since for an A containing A as a J -array where AA ■ B(u) T s s and A A * B(w), we must have by Corollaries 3»h, 3.5* and 3.6 that T -k jki u A ■ e K w and w A m t k u. whence u A m e k w and s s -*> ArT -ki w A m c k u where e » 1. -1. r This theorem yields as a corollary the following result which is of interest in the study of v, k, X designs. Corollary U.5. Let B = B("f) « I. Suppose A^ is a f- r -array, 0 < r, s < v, composed of 0 's and l's such that s ' Ar ArT -B(t*) and Ag A s - Btf*). Then there exists a v x v r — r rational ttairix A containing Ag as a j -array and satisfy- ing AAT - ATA - B. Proof: The conditions for the rational completability of r ^ i r -array to the desired matrix are that 1 A ■ e H 1 and *?Ar^ ■ c k where e - 1, -1. Now by ArArT m and T -*i r A^ A ■ B(T^) every row of A and every column of A^ has exactly k l's and v - k 0's. Hence the conditions are satisfied. Ill* Integral Solutions to the Incidence Equation: for Finite Projective Plane Cases of Orders n ~ 2 (mod U) 1. Introduction. In [10] Ryser proved that any normal inte gral solution to the incidence equation must either be a design matrix or the negative of one. He further proved there, in "his theorem 2,2 which has been mentioned earlier, that for k - X odd and gcd (k, k-X) • 1, any integral solution to the incidence equation (normality not assumed) must either be a design matrix or a matrix transformable into one by suitable multiplication of its columns by - 1* We notice that this theorem is applicable to fin ite projective plane cases of odd order and to Hadamard design cases of order lun-1 where m is odd. Actually this theorem is still true with the condition gcd (k, k-X) • 1 replaced by the condition that the gcd (k, k-X) is squarefree, but here we will assume only the former condition. This theorem is not true, how ever, when k - X is even. When k - X is even, more exotic situations may and do occur. We may have design type integral solutions like those above for k - X odd which we shall call type I solutions, or we may have integral solutions which are not of that type which we shall call type II solutions. In the Appendix we discuss some results on the integral solutions to the incidence equation for general values of v, k, X under the con- dition that gcd (k, k-X) • 1, and using these as aids construct examples of type II' solutions for small values of v. Here, how ever, we will be solely concerned with type II solutions for the finite projective plane case for which we here let n denote the 9 order. In [10] Ryser shows that a type II solution exists whenever n - 2 or n = 0 (mod h) is also the order of a Hadamard matrix, and in [7] Hall and Ryser exhibit such a solution for n » 10* Here we shall construct type II solutions for some infinite classes of val ues of n = 2 (mod li). By section L of [7] we know that we can put any type II solution A • [aijl into a canonical form where a^ j ■ 0, a n " ^ ^or 1 < i < v, , ■ 1 for J 5 2 (mod n) and a^j • 0 for J ^ 2 (mod n) where 1 < j < v, and the remaining entries form a submatrix C of order v - 1 ■ n(n+l) which has n l's and the remaining entries 0 's in each of the n + 1 col umns under a 1 in row 1 of A and which satisfies the matrix T T equation CC ■ C C ■ n I. The constructions given by Ryser in [1G] and Hall and Ryser in [7] have C in the form C - An © ... © A^ , where the direct sum contains AR , of order n, n + 1 times and where ^ has all entries in column 1 equal to 1 and satisfies the T matrix equation A^A^ » n I. These conditions on A^ are sufficient for the construction of a type II solution for order n. We shall confine ourselves here to this form of type II solution. This restriction reduces the construction of a type II solution A 2 of order n + n + 1 to that of an integral matrix An of order n satisfying the above conditions. Type II solutions need not, however, be of this direct sum form within permutation of rows and columns of A. This can be seen from the following example for 25 n ■ lw 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0* 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 0 0 1 -1 1 1 - 1 0 0 -1 I 1 1 1 1 1 0 0 1 -1 -1 1 1 0 0 1 1 -1 0 0 1 -1 1 1 -1 0 0 -1 1 1 1 1 1 1 0 0 1 -1 -1 1 1 0 0 1 1-1 0 0 1 -1 1 1-1 0 0 -1 1 1 1 1 1 1 0 0 1 -1 -1 1 1 0 0 1 1 -1 0 0 1 -1 1 1 -1 0 0 -1 1 1 1 1 1 1 0 0 1 - 1 --1 1 1 0 0 1 1 -1 1 1 1 1 -1 -1 -1 b Basic to all the constructions of type II solutions presented here are the skew-iiadamard designs whose orders satisfy certain in tegral equations. These designs form the skeleton upon which the integral matrices are constructed. We, of course, construct type II solutions only for values of n = 2 (mod U) which satisfy the Bruck-Ryser criterion (I, Theorem 1.1), In particular, this criterion rules out the existence of rational hence integral solutions to the incidence equation for all orders n = 6 (mod 8) along with some orders n ~ 2 (mod 8). This criterion is equiv- 2 2 alent to saying that n » a + b where a and b are odd in tegers. For certain subclasses of the classes of orders 2 2 2 2 n ■ a + 1 , 2a,a + (a + 2) , where a is odd, we obtain con- struction theorems for type II solutions* 2. Skew-Hadamard Designs and Matrices. Let H « hrs « 1, -1, be a matrix of order n. We call H a Hadamard matrix if HH « n I, By an inequality of Hadamard [6 ] this latter condition is equivalent to saying that jdet h | ■ n ^* We immediately see T also that H H ■ n I, It is easy to show that if a Hadamard matrix exists then its order n must be 1, 2 or a multiple of L, and that the direct product of two Hadamard matrices is a Hadamard matrix, i,e., that if there exist Hadamard matrices of orders m and n, then there exists one of order m n (on both points see [6 ] and [3]). Several people have constructed classes of Hadamard matrices, many of them being infinite; see [2 ], [6 ], [8 ], [11], [12], [la], and [1J>]. As mentioned in part I there is a direct connection be tween Hadamard matrices and a certain class of v, k, X designs called Hadamard designs, which is given by the following result proved by Todd [13]. Theorem 2*1. A Hadamard matrix of order n - Um exists if and only if a v, k, X design with parameters v » hm - 1, k - 2m - 1, and X * m - 1 exists. Proof: Let H » [h ] be a Hadamard matrix of order rs n » Lm. By multiplying the appropriate columns and rows by -1 we can convert H to a Hadamard matrix H* ■ [h* ] where h?. - 1, rs ij 27 (2.1) H* - Any two rows of H" can be schematically represented as follows: • • • i lj lj i • • j lj “ 1 j ••• _ 1| “ 1) •••) *• 1 ,1> If ...» lj -1| . M j -1 1*1 » I <• ] d I Now from (2,1) and (2,2) we have a + b + c + d«i*m-l, a + b » 2m, a + c • 2m, and a-b-c+d- -1, which yields a » b - c ■ m (2.3) d • m - 1 . We now substitute a 0 for each 1 and a 1 for each -1 in to obtain a 0, 1 matrix A of order v - Ijm - 1 which by (2*2) and (2.3) satisfies (2.h) AAT - m I + (m-l) S, S - [s.s.. - 1 , ij iJ Hence A is the incidence matrix of a v, k, X design with parameters v • bn - 1, k - 2m - 1, and X ■ m - 1, Clearly the argument is reversible whence the converse is also true. T We call a Hadamard matrix H skew-Hadamard if H + H • 2 I. These also exist in infinite classes as will be shown later. We T also call a Hadamard design A skew-Hadamard if A + A • S - I, The reason for the agreement of the descriptions in these two definitions will be clear from the following lemma. Lemma 2.2. To a skew-Hadamard matrix of order n » lun there corresponds a skew-Hadamard design with parameters v - Lm - 1, k • 2m - 1, and X - m - 1, and conversely. Proof: By examining the argument in Theorem 2.1 we see that if ik Ik Ik H is skew-Hadamard then h ■ 1 and h. - -h * l (2.1). For n - 1 the matrix [1] is trivially a skew-Hadamard; its corresponding design, being of order 0, is vacuous. For n • 2 the matrix [-! i] is skew-Hadamard; its corresponding design is the matrix [1] of order 1 which is trivially skew-Hadamard. Among the matrices of order n » lm with entries 1 and -1 we can characterize those that are skew-Hadamard by the following theorem. Theorem 2.3* Let H - [h ].h -1,-1, be a matrix of — — rs * rs — ---- T order n - Lim and let G-H+H - 2 1. Then the following statements are equivalent; (a) H is a skew-Hadamard matrix. 29 (c) The eigenvalues of H are 1 + i V n“l and 1 - i each with nrultipliclty 2m. 2 (d) H is a Hadamard matrix and trace G ■ 0. Proof i Me shall show that (a) implies (b) implies (c) implies (a) and that (a) implies (d) implies (a). Let H be a skew-Hadamard matrix. Then HHT - n I and T H + H - 2 1 imply (b). Now suppose that (b) holds. Since H cannot satisfy a 2 first degree matric equation, X - 2 X + n must be its minimal polynomial whence only 1 + iVn-1 and 1 - i V n_l are its eigen values. Now the trace of H is real; hence the two complex eigen values must occur with the same multiplicity, namely, 2m. Now assume that (c) holds. Then 2m 2m § det H ■ (l + i Vn-1) (1 - i Vn-1) ■ n , 2 whence H is a Hadamard matrix. Since the eigenvalues of H are 2 - n ♦ 2 i Vn-i" and 2 - n - 2 each with multipli city 2m, we have moreover that 2 2 2 2 (2.5) trace (H - 2H + nl) ■ trace(H^- 2 trace (i^-n • -Lmn+n •» 0, Since h * 1, -1, rs ' n (2.6) V " -1 ^ h . - 2h + n > -(n-2) -2 + n ■ 0, 1 < r < n. / ri lr rr — — — i-i 30 Now equality must hold in (2.6),else we contradict (2.5). But if equality holds in (2.6) then hr - h ir • -1 and h^^ « 1, 1 < i, r < n, i ^ r. This implies that H is a skew-Hadamard matrix. T 2 Finally if (a) holds then G - H + H -21-0 whence trace G - 0. 2 Suppose that (d) holds. Since G is symmetric, trace G - 0 implies that the sum of the squares of the elements of G is 0. Hence G » 0 and H is a skew-Hadamard matrix. We may also inquire as to whether there is a direct product type of construction for skew-Hadamard matrices. Such a result can be obtained as a corollary to the followinp lemma of Williamson [lli] in which I denotes the identity matrix of order r and x de- r notes direct product. Lemma 2.L. Let C be a matrix of order n such that T T C « e C, c » 1, -1, and CC « (n-1) In, and let D and E be T T ------two matrices of order ------m satisfy!no--— DD » EE - m l m --- and T T DE - -eED . Then the matrix R-'DxIn + E x C satisfies T KK - r a n I . mn The corollary of interest to us here for skew-Kadarard matrices is the following one. Corollary 2.5. Let C + I be a skew-nadamard matrix of order n, and let Q be a skew-Hadam.ard and E a symmetric T T Hadamard matrix of order m such that DE - ED . Then the matrix K-DxI^ + ExC j_sa skew-Hadamard matrix of order m n. T Proof: Since C + I is a skew-Hadamard matrix, C - -C T and CC - (n-1)I , and since D and E are both Hadamard n 31 T T I T matrices* DD » EE -ml. Now e - -1 and we have DE m ED . m Finally since D is skew-Hadamard and E is symmetric: T T K + K «DxIn +ExC+(DxIn+ExC) T T T » D x L + E x C + D x I + E xC n n - (D + DT) xIn +ExC-ExC • 2 Lm x I n -21 , mn and we have the corollary by Lemma 2.b. Finally we give a theorem which states the existence of skew- Hadamard matrices for certain classes of the order n. These re sults were essentially proved by Williamson [lb] who obtained some of the special cases from the work of Paley [8], Theorem 2.6. There exist skew-Hadamard matrices of order n where n rs any of the following forms: (a) 2°, c > 0 . r hi hi - (b) 'jy (p. + 1), r > 0 and p. + 1 ; 0(mod b), p. a prime, i-1 1 1 1 < i < r. r hi , hi t o 2C TT (p.tF + 1), c, r > 0‘ and p^ +1-0 (mod b), i-1 Pi £ prime, 1 < 1 < r. Table 1 gives the known existence of skew-Hadamard matrices for orders b < n < 200 according to Theorem 2.6. For comparison the table also gives the known existence of Hadamard matrices for the same range of n based on constructions in the references mentioned earlier. The symbol SH indicates that a skew-Hadamard matrix is known to exist while the symbol h indicates that only a nonskew- Hadamard matrix is known to exist. 33 Tabic 1 The Existence of Hadamard and Skew-Hadamard Matrices for Orders U < n < 200 n Form There Exists n Form There Exists 2 k 2- SH 10b 103+1 SH 8 2 SH 108 J07±l SH 12 11+1 SH 112 22 (33+l) SH 16 24 SH - 116 20 19+1 SH 120 2(59+1) SH 2k 2(11+1) SH 12b 7 h SH 128 2 SH 28 3 V 32 23 SH 132 131+1 SH 36 h 136 2(67+1) SH iiO 2(19+1) SH iko 139+1 SH kk _U3+1 SH lbb 2(71+1) SH L8 2(11+1) SH 1L8 h 52 h 152 151+1 SH 56 2(3+1) SH 156 'I 60 59*1 SH 160 23 (19+1) SH 6k 2 SH 16a 163+1 SH 68 67+1 SH 168 2(83+1) SH 72 71+1 SH 172 0 h 76 h 176 2 (U3+1) SH 80 22 (19+1) SH 180 179+1 SH 8b 83+1 SH 18b h* 88 2(b3+l) SH* 188 t, 92 h* 192 2(11+1) SH 96 2(11+1) SH 196 h 100 h 200 199+1 SH * A recent communication from Marshall Hall asserts that a Hadamard matrix of order 92 has been found by Leonard Baumert using the IBM-7090 computer. By a direct product construction, one of order 18b then also exists. 3. Construction Theorems. Let K be a skew-Hadamard design matrix of order q = 3 (mod U). Here v • q » ljm -1, k • 2m -1, X ■ m - 1, where m > 0 is an integer, (3.1) KKT - KTK - m I + (m - 1) S, 4 and (3.2) K + K? - S - I, where S»(s.j], Sj j - 1, l onal 0's, u for each of the remaining 0's and x for each of the l's. From (3.1) and (3.2) any two rows of K(t, u, x) can be schematically represented in columns as t u u ... u u ...ux ... X X ... X ^ ^ X t X ... XU... UX... XU... u |1^1|m-l|m-l|m-lj m | where there are hm -1 entries in each row, 2m-1 each of u's and x's. The inner product of"a row of K(t, u, x)_ with itself is thus (3.U) t2 + (2m - l)(x^ + u2) • t2 + ^ (q - l)(x2 + u2) , Also the inner product of two distinct rows of K(t, u, x) is (3.5) t(x + u) + (m - l)(x2 + u2) + (2m - l) x u 1 2 2 1 » t(x + u) + £ (q -3)(x + u ) + - (q - 1) x u 1 2 1 2 2 • t(x + u) + - (q - l)(x + u) - J (x + u ). We now form Y » ^ijl " K(t^, u^, x^) and Z - ^zij^ “ K(t^, u2, x ) of order q and then form Lemma 3.1. The matrix equation (3.7) NNT - w I is satisfied if and only if 2 ? I, .,,2 2 2 2* - (3.8) + t 1 2 2 + 2 2q " 1 + U1 + X2 + V : ■ ( h + |(q - !)(*! + u 2)]2+ tt2 + |(q - l)(x2+u2)]2. Proof: By (3.6) we have T T YY +ZZ, 2Y-YZ (3.9) m (Z Y - Y Z)T , YTY + ZTZ Since, by (3.1), K is a normal matrix, the statements about inner product values of K(t, u, x) are true when the word row(s) is replaced by column(s); hence K(t, u, x) is normal whence Y and Z are normal or (3.10) YTY - YYT , ZTZ - ZZT . Now •Y - tj I + x K + u 1 (S - K) _-u 1 - (tj - u1) I + (x1 - u ^ K + u1 S , and similarly Z • (t2 - u2) I + (x2 - u?) K + u^ S . Since I commutes with both K and S and 36 K S ■ S K • (2m - 1) S , i.e., K commutes with S, then Y commutes with Z or (3.11) Z Y - Y Z - 0 . Then by (3.10) and (3.11)j (3.9) becomes (3.12) NNT - (YYT + ZZT ) © (YYT + zzT ). T The diagonal entries of NN are by (3.1i) and (3.12) ... .2 2 1, 2 2 2 2 (3.13) t2 + t2 + -(q - l)(x1 + u1 + x? + u ) - w, and the nondiaaonal entries of the direct summands in (3.12) are by (3.5) 1 2 2 (3. Ill) t^Xj + ux) + t2 (x2 + u2)+ ^(q - l)[(x1+ ux) +(x2+ u2) ] 1 , 2 2 2 2. “ 2 (X1 + U 1 + X2 + U2 " y * We note that (3.7) is satisfied if and only if y » 0. Now 1 , 2 d 2 2d 2 d solving (3. Ill) for — (x. + u + xt + u ) and substituting the 2 1 1 22 2 results into (3.13) we obtain '(3.15) [tL+ |(q - l)(xj+ u1)]2+.[t2+ ” (q -l)(x2+ u2)]2-(q -l)y-w. Hence by (3.13)> (3*lli)» and (3.15)> we see that (3.7) is true if and only if (3.8) is. We now define the matrices Er • -^(r + 2)1 - S of even order r, Fr of size r x 2 consisting entirely of l's and G^. of size r x 2 whose first column consists entirely of l's and whose second column consists entirely of -l's. In the con structions which follow we shall be taking t^ - -|(r + 2) and X1 + U 1 " ^ cn n°te that 37 (3.16) F F 1 + E ET - G GT + E ET • [^(r + 2)]2 I - t2 I , ' ' r r rr rr r r 2 1 (3.17) FrFr + 2Er - GpOj + 2Er * (r + 2) I ■ ( x ^ u ^ I , and (3.18) F G ^ » G F^ ■ 0 , ' * r r r r T We substitute for the entries y.. in Y and Y the matrix J i i Ef and for all other entries Y j j # i / j> the matrix yjjl °f T order r to obtain the matrices Y# and Y , respectively, of T order r q, and substitute for the entries z. . in 2 and 2 ij T the matrix z.i j .1 of order r to obtain the matrices 2„ * and ZM , respectively, also of order r q. These matrices will appear in the constructions which follow, bordered by the matrices Fr<^ and V We can now obtain two existence theorems for.type II sol utions to the incidence equation for finite projective plane cases of orders n = 2 (mod L). After each one are theorems which cover the various cases of the theorem. Theorem 3.2. Let (3.8) be satisfied in integers t p t p U p u2> x^, x2> and w, where t^ * -|(r + 2), x^ + u^ - .2, and w - 2rq + 2. Then there exists a type II solution to the incid ence equation for the finite projective plane case of order n « w - 2rq + 2.- Proof: We have N 38 where (3.19) y u * tl - |( r + 2), y.^+ y^.- 2, i ^ j, 1 < i, j < q> and (3.20) M T - (2rq + 2) I . Since (3.8) is satisfied we have [ | ( r + 2) + (q - 1)]2+ [t2+ |(q - l)(x2+ ■ 2rq + 2, or U - |r]2 + [tg + j (q - l)(x2 + u2)]2 - 2 . 1 1. , Since q, “ r, V 2 q ' Xg, and are integers this means that (3.21) q - ~ r - tg+ |(q-l)(x2+ ug) - e2 , - 1, -1 , We form two matrices U and V of size 2 x r q according to the values of and as follows: U V (3.22) if e, - -e - 1. t L 1 2 if e, ■ -e » -1. [ [■ 1 2 39 Finally we construct A^ of order n » w ■ 2rq + 2: (3.2 3) A, m 3y (3.22) and (3.23) the first two rcws of A^ are mutually orthop- onal and have self inner products equal to n ■ 2rq + 2. Since the rov and column sums of Y are q - —r and those of Z are # n 2 * t2 + 2^q " 1^ x2 + u2‘>* we haVC by (3.21), (3.22), and (3.23) that rows one and two are orthogonal to all the other rows of An» We T may now verify that A A « n 1 by lookina at the naturally de- n n fined blocks of r rows below the second row of A • Since the n terms in t^ in the entries of (3.11) cancel to 0 identically, we have by (3.18) and (3.20) that in (3.23) any rcw block inter- sectinc F_„rq is orthogonal - to anyj row block . intersectinc » G rq . Finally, by (3.16), (3.17), (3.19), and (3.20), any row block below row two of A^ has self inner product equal to (2rq+2)l ■ n I of order r. Notino that the first column of A has all entries n equal to 1, we see that we have a type II solution to the incidence equation for the finite projective plane case of order n m 2rq-t2. Letting c - and combining (3.21) with the first part, of (3.8), noting that ^ (r+2) ■ q - Cj + 1, we have Uo [q-e1+ 1] + U 2- |(q-l)c] 2 + |(q-l) 1/ vr[x 2 ^ ^ / - x ^ 2+ x^tc-x^) 2 ] « aq^q-e^ + 2 , or (3.2li) -62c(q-l)+ ±c2(q-l)h |(q-l)[2(xr l)2f2Cx2- |c)h \ c h 2] 3q - 2tj q ♦ 26j - 2q-l C3q - (2e1 - l)](q-l), or 1 2 , 2 1 2 1 2 -e2 c + £ c (q-l)+(x1- 1) + (x^- “ c) + £ c + 1 - 3q - 2e1 + 1 , whence (3.25) (12 - c2)q + Le2c~- ce - (2x*1~ 2)2 + (2x2~ c)2. Ey (3.25) (12 - c^)q + lie2 c - fie^ > 0 and since q > 3> (*\ r2 ^£2 U ^ei L t3.2oJ c ----- — c + “7 < 12 + _ < ---136 q q — q q2 “ 9 Since c is an integer we can readily conclude that (3.27) | c| < I . We let a = 2x^ - 2 and b = 2x2 - c. Since q = Itm - 1, where m > 0 is an integer, we have from (3.25) that (3.28) (12 - c 2 )().!ir. - 1) + hc2 c - 8e^ ■ a^ + b^. By (3.27) we have nine values of e^c for each of the two cases corresponding to = l'» ~lJ hence we have a total of eighteen cases to consider in which the left side of (3.28) is equal to the sum of two integral squares. We take the nine cases for £ 2 * 1 * 2 2 Case 1. c^c » 1; : -16m + 12 ■ a + b , impossible since -l6m + 12 < 0 for m > 0. 2 2 Case 2. e£C « 3 : 12m + 1 * a + b , possible since e.g., 12(l) + 1 ■ 13 ** 32 + 22. Here 3q + k * a2+ b4- * Case 3. &2C " ^ 1 8(Lm - 1) ■ a2 + b2 or hm - 1 m *a2 + b2, a^, b^ intecers, impossible since hm - 1 ^ 3(mod 14). 2 2 Case L. e^c • 1 : LLm - 15 ■ a + b , possible since e.c,, Lii (l)— 15 - 29 - 52 + 22. Here llq - 1* - a2 + b2. 2 2 2 2 Case 5, e^c ■ 0 : k8m - 20 • a + b or 12m - 5 * a^+ b^ , a^, bj integers, impossible since 12m-5 « 3 (mod L). 2 2 Case 6. e„c ■ -1 : Linn - 23 » a + b , possible since e.g. , Lh(2)- 23 - 65 - 82 + l2. Here llq - 12 - a2+ b 2. 2 2 2 2 Case 7. &2C “ ”2 : 32m - 2k • a + b or kn - 3 « + b^ , a^, b^ integers, possible since e.g. , k(2)-3 ■ 5 » 22+ l2. Here 8q - 16 ■ a2+ b2 or 2 2 q - 2 - a + b . 2 2 Case 8. t^c « -3 : 12m - 23 ■ a + b , possible since e.g. , 12(3)- 23 - 13 - 32+ 22. here 3q- 20 « a2+ b2. 2 2 Case 9. £2° ” : -16m - 20 » a + b , impossible since -16m - 20 <'0 for m > 0, Clearly, given a possible case where a and b are integers, we can reverse the argument to determine an integral x^ and 12 whence also an integral u^, u^, and t^y which together with t^ - “ (r+2) and w • 2rq+2 furnish a solution in integers to (3.8), 2 2 Now when « 1, r « 2(q-l), hence n ■ Ltq - Lq ■* 2 ■ (2q-l) + 1. So by Theorem 3.2 we have the following result. Theorem 3.3. There exists a type II solution to the incid ence equation for the finite projective plane case of order 2 n ■ (2q-l) + 1 whenever q _is the order of a skew-Hadamard design matrix and any of the following express ions is the sum of two in- tegral squares: 3q+L * llq-L, llq-12, q-2, 3q-20. 2 When • -1, r ■ 2(q+l) hence n « Lq + Lq + 2 » 2 (2q+l) + 1. Analyzing this case as was done above for » 1, we have by Theorem 3*2 the corresponding result: Theorem 3.U. There exists a type II integral solution to the incidence equation for the finite projective plane case of 2 order n » (2q+l) + 1 whenever q ^ the order of a skew-Hadamard design matrix and any of the following expressions is the sum of two integral squares: 3q-L, llq+L* llq+12, q+2, 3q+20. Both of these theorems yield infinitely many type II solut ions. There exist skew-Hadamard design matrices of orders qx - 2^d“2(ll+l)--1 - 3*22d-l and q2 - 22d~2(L3+l)-l - U * 2 2d-1 d 2 2 where d > 1 is an integer. Then 3q^+ L - (3*2 ) + 1 , and d 2 2 llq + 12 ■ (11*2 ) + 1 . The first five orders for which Theorem 3.3 yields solutions correspond to q*3, 7, 11, 15, 19 and are n » 26, 170, LL2, 8L2, 1370, respectively. Likewise for Theorem 3.L, q « 3, 7, 11, 15, 19 and n - 50, 226, 530, 962, 1522, h3 respectively. As an example we construct A ^ . For n - 26 we have q ■ 3 and • 1 hence r » h whence t^ ■ 3. Wow by case 2 above, e^c m- 3 and 3q + I - 13 - 22+ 32 - (2xx- 2)2 + (2x2~ c)2. We take 2x^- 2 » 2 or Xj» 2 and 2x^- c » 3. Letting - 1, we have c ■ 3 whence x2 ■ 3 and t^- -2. Then • 0. Mow E - 31 - S of order U and since e, ■ e - 1, i_l i d T-i . . . - f '-1 . . . -1 u - and V • L i . . . i -1 . . . -1_ o f size 2 x 12. The matrices F, and G, are of size U x 2 a a and a skew-Hadamard design matrix of order 3 is "o 1 o" 0 0 1 _1 ° °_ Hence we have "l 1 -1 * ♦ • -1 -1 • • • -1 1 -1 1 • « • 1 -1 • » • -1 1 1 3I-S 21 0 -21 31 0 ■ 0 3I-S 21 0 -21 31 • 1 1 21 0 3I-S 31 0 -21 1 -1 21 0 -31 3I-S 0 21 • • “3I 21 0 21 3I-S 0 • J -1 0 -31 21 0 21 3I-S The second existence theorem for type II solutions is the following one. hk Theorem 3.5. Let (3*8) be satisfied in integers t^, t^, u^, Ug, Xj,, and w, where t^ m ^(r+2), x^ + • 2, and w ■ 2rq ♦ 1. Then there exists a type II solution to the incidence equation for the finite projective plane case of order n - 2v • Urq ♦ 2. Proof: We have Y Z N T T -Z Y where (3.29) y u * tj- |(r+2), y.j+ y ^ * Xj+ I, I ^ j, 1 < i, j < q, and T (3.30) NN - (2rq + 1)1 . Since (3.8) is satisfied we have [“(r4-2) + (q-l)]2+ [t2+ |(q-l)(x2+ u^))2 m 2rq + 1, or 2 tq - | r]2 + [t2+ |(q-l)(x2+ u2)] •» 1. Since q, ^ r, t^t ^(q-l), x^» and u2 are integers this means that (3.31) q - | r « e1, t2+ |(q-l)(x2+ u2) - 2 2 + ^2 ■ lj s2 » 1^ 0| -1. We form two matrices U and V of size 2xrq according to the values of and e2 as follows: h $ u V (3.32) f-2 0 if ■ 1» ®2 *" t [-2 :l] [ 3 [:: if - -1, e2 - 0. [ t:: if - 0, e2 » 1. 0 if £ “ 0, ^2 m [-2 :] [:: We set and 9 - t2 + j (q-l)(x2+ u?) - e , Then f and g are integers and by (3.6) 2 2 (3.33) f + g ■ w » 2rq + 1 . Finally we construct A^ of order n - 2w • lirq + 2: "l 1 U V 0 0 1 -1 Y z fl F rq * * rq ^ r q F Y z -gi rq . * tt -fIrq rq (3.31*) A G y t -fl rq * 9lrq rq T -gl < Y* a rq rq - G r q U6 By (3*32) and (3.3U) the first two rows of Aft are mutually ortho gonal and have self inner products equal to n ■ Urq + 2. Since the row and column sums of Y~ are q - — r and those of Z.t are 2 t2+ 2^q-1^ X2+ We haVe by (3-31)» (3.32), and (3.3U) that rows one and two are orthogonal to all the other rows of A . We may T again verify that * n I by looking at the naturally defined blocks of r rows below the second row of A . Since the terms in n in the entries of (3.11) cancel to 0 identically we have by (3.18), (3.30), and (3.3U) that any row block intersectinci an F rq is orthogonal to any row block intersecting a ^rq* Finally, by (3.16), (3.17), (3.29), (3.30)', (3.33), and (3.3U), any-two dis tinct row blocks intersecting either an F or a G are rq rq mutually orthogonal, and any row block below row two of A has n self inner product equal to (8rq + 2)1 » n I of order r. Not ing that the first column of An has all entries equal to 1, we see that we have a type II solution to the incidence equation for the finite projective plane case of order n - lirq + 2. Letting c - x^+ u^ and combining (3.31) with the first part of (3.8), noting that t1 - ^(r+2) - q - e^+ 1, we have [q-e + 1]2+ [ert- ^(q-l)c]2+ 7 (q-l) [x^+ (2-x )2+ x^+ (c-x )2 ] 2q*2(q - + 1, or 17 -e2c(q-l)+ £ c2 (q-l)2+ |(q-l)[2(x1-l)2+ 2 ^ - |c) + | c2+ 2] 2 * 3q - 2e1 q ♦ 2e^ - 2q - 1 - [3q - (2c^- 1)] (q - 1) which is identical to (3.2l*). Hence we obtain (3.25) and (3.26). Since the argument frcm (3.26) to (3.27) depends only on |£l| * |C2 1 — * anc* c*' — ^ and s^nce this is true here too, we obtain (3.27). Again, letting a ■ 2x - 2, b » 2x?- c, and q ■ km - 1, m > 0 an integer, we obtain as before 2 2 2 (3.28) (12 - c ) (km - 1) + Le^ c - 8e^ ■ a + b , where (3.27) < li . By (3.27) we have five values of jc| for each of the two cases e, ■ 1, e - 0 and e, ■ -1, e„ • 0 and nine values of e„c for 1 2 1 2 2 the case ■ 0, hence we have a total of nineteen cases to con sider in which the left side of (3.28) is equal to the sum of two integral squares. We take the five cases for * 1, » 0. II 2 2 Case 1. [cl « U: -16m - k ■ a + b , impossible since -16m - k < 0 for m > 0. Case 2. | c | - 3: 12m - 11 ■ a2+ b , possible since e.c. , 12(2)- 11 - 13 - 32+ 22. Here 3q - 8 - a2+ b2. i l 2 2 2 2 Case 3. | c | - 2: 32m - 16 ■ a + b or 2m - 1 • a^+ b^, a integers, possible since e.g.,2(3)-l ■ 5 ■ 2 2 2 2 2 2 2 + 1 . Here 8q-8 - a + b or q-1 ■ a + b , 2 2 32 ’ ^2 ^tecers. 1*8 I I 2 2 Case U. | c I - 1: liUm - 19 • a + b , possible since e.g., 2 2 2 2 ill*(1) - 19 - 2$ - 5 + 0 . Here llq-8 » a + b . I 2 2 2 2 Case 5. | c | - 0: ii8m - 20 » a + b or 12m - 5 • b i* a^, b^ integers, impossible since 12m-5 ~ 3 (mod'ii). Again, given a possible case where a and b are integers, we can reverse the argument to determine an integral x and x whence 1 2 also an integral u^, u^, and t^, which together with t^* ” (r+2) and w ■» 2rq+l furnish a solution in integers to (3,8). Wow when 2 2 e » 1, r • 2(q-l), hence n » 8q - 8q + 2 « 2(2q-l) . So by Theorem 3.5 we have thefollowing result. Theorem 3.6. There exists a type II solution to the incid ence equation for the f inite projective plane case of order 2 n - 2(2q-l) whenever q j_s the order of a skew-Hadamard design matrix and any of the following expressions is the sum of two in- tegral squares: 3q - 8, q - 1, llq - 8. 2 2 When e1 * -1, r ■ 2(q+l), hence n - 8q + 8q + 2 Analyzing this case as was done above for - 1, we have by Theorem 3.5 the corresponding result: Theorem 3.7. There exists a type II solution to the in cidence equation for the finite projective plane case of order 2 n ■ 2(2q+l) whenever q ^s the order of a skew-Hadamard design matrix and any of the following expressions is the sum of two in tegral squares: 3q+8, q+1, llq+8. 2 2 2 When 0, r ■ 2q, hence n « 8q + 2 » (2q-l) + (2q+l) . Analyzing this case as was done for Theorem 3.3 we have by Theorem 3*5 the following result. Theorem 3.8* There exists a type II solution to the incidence equation for the finite projective plane case of order 2 2 ■ n ■ (2q-l) + (2q+l) whenever q i_s the order of a skew-Hadamard design matrix and any of the following expressions is the sum of two integral squares: 3q + 12, q + 1, llq + lj, 3q, llq - L, q - 1, 3q - 12. All three theorems yield infinitely many type II solutions. There exist skew-Hadamard design matrices of orders q^ « L ( 3 ^ “^+ l)-l ■ li*32d 3 and q^« 22c*-l where d > 1 is an d 2 2 2d 2 integer. Then 3q^- 8 * (2*3 ) + 1 , and q^+ 1 * 2 + 0 . The first four values for which Theorem 3.6 yields solutions correspond to q ■ 3, 7, 11, 15 and are n ■ 50, 338, 882, 1682, res pectively. Likewise for Theorem 3.7, q * 3, 7, 11, 15 and n ■ 98, L50, 1058, 1922, respectively, and for Theorem 3.8, q * 3, 7, 11, 15 and n - 7lj, 39L, 970, 1802, respectively. ’ As an example we construct A^q. For n » 50 we have q ■ 3, » 1, and * 0 hence r » L whence t^ ■ 3. Now by case it above, J c | » 1 and llq - 8 • 25 - 0^ + 52 » (2x^- 2)2 + (2x2~ c )2 . We take 2x^- 2 • 0 or x^ ■ 1 and 2x^ - c - 5. Letting c - 1 we have x^ • 3 and t£ • -1. Then » 1 and u ■ -2, f - 5 and g m 0. Now • 31 - S of order It and since - 1 and 5 0 i o *-2 . . . ~2 • • • u - and V ■ o o • • * -2 . . . -2_ 1 of size 2 x 12. The matrices F. and G, are of size h x 2, and u « a skew-Hadamard desiqn matrix of order 3 is 0 1 0 0 0 1 • 1 0 0 Hence we have 1 1 -2 • • * -2 0 • • • 0 0 0 1 -1 0 * • « 0 -2 . • • -2 1 1 3I-S I I -I 31 -21 51 • • I 3I-S I -21 -I 31 51 0 « 1 1 I I 3I-S 31 -21 -I ' ■ 51 1 1 3 I-S I I -I 31 -21 -51 * I 3I-S I -21 -I 31 -51 0 • 1 1 I I 3I-S 31 -21 -I -51 “ 1 -1 I 21 -31 31-3 I I -51 * -31 I 21 I 3I-S I 0 -51 • 1 -1 21 -31 I 1 I 3I-S -51 1 -1 I 21 -31 3I-S II 51 • • -31 I 21 I 3I-S I .0 51 _l -1 21 -3! I 31-3 II 5I- These theorems are all based on the existence of a skew-Hada- mard design matrix of a certain order q ■ 3 (mod lj). Their proofs become uniform, under this restriction, however, a trivial skew- Hadamard design exists for order q • 1 which corresponds to a skew- Hadamard matrix of order 2. Here we have N - , of order 2, -t. where (3.35) NN1 w I if and only if 2 2 (3.36) tl + l2 = W * Let us consider the form of construction in Theorem 3.2. We let (3.36) be satisfied in integers t^ « ^(r+2), t,,, and w - 2r+2. Then * r (r+2)2 ■+ t2 - 2r + 2, U 2 or 1 2 2 r ♦ t2 - 2 , hence 1 " 2 r “ V t2 " V V £2 " lj 'L For e ■ 1, r » 0, hence we pet no nontrivial construction. For ■ -1 we obtain r ■ L whence n ■ w m 10. We have » 31 - S of order L and F^ and G^, as defined previously, of size L x 2. Then corresponding to - 1, -1 we obtain by the construction in Theorem 3*2 52 3I-S 3I-S -I 9 3I-S 3I-S respectively, which are virtually the same as the constructed by Hall and Ryser [7]. Now let us consider the form of construction in Theorem 3*5. We let (3.36) be satisfied in integers t^* “ (r+2), t£, and w ■ 2r+l. Then l 7 7 — (r+2) + t - 2r+l, h 2 or 1, .2 2 r(r-2) + t - 1, L 2 hence 1 2 2 1 " " tl* l2 “ V £1+ % " lf V C2 " 1j °* ”1' For ■ 1 we aaain get no nontrivial construction. For - 0 we obtain r * 2 whence n ■ 2w » 10. We have E » 21 - S of 2 order 2 and F,, and 0^, as defined previously of size 2x2. Then corresponding to ■ 1, -1, f « 2, and g - 1, -1, res pectively, and we obtain by the construction in Theorem 3.5 53 ’l 1 0 0 -2 -2 "l 1 0 0 2 2 0 ow 1 -1 2 2 0 0 1 — 1 -2 -2 0 0 1 1 1 -1 1 0 2 0 1 0 1 1 1 -1 -1 0 2 0 -1 0 CM o 1 1 -1 1 0 1 0 1 1 1 -1 1 0 -1 0 2 0 -1 1 1 1 -1 i 0 -2 0 -1 0 1 1 1 -1 -1 0 -2 0 1 0 Aio" 1 1 -1 1 0 1 0 -2 0 -1 » 1 1 -1 1 0 -1 0 -2 0 1 O C m 1 -1 -1 0 i -1 1 1 0 1 -1 1 0 1 -1 -1 0 -2 0 1 -1 0 -1 -i 1 0 1 0 -2 1 -1 0 1 -1 1 0 -1 0 -2 1 -1 -1 0 l -1 -1 0 2 0 1 -1 1 0 1 -1 1 0 2 0 1 -1 0 -1 -l 1 0 -1 0 2^ 1 -1 0 1 -1 1 0 1 0 2 resnectively,J which a-e essentiallyJ different from the ones for A,„ jr, previously exhibited. This shows that even type II solutions of the direct sun tyoe are act necessarily unique to within permuta tions of rows and columns and their multiplication by 1, -1, Finally, for » -1, » 0, we obtain r • 1* whence n - 2w » 18. he ' / . y >c a. » 31-5 of order' L a.id r. and G. , as previouslv I k h defined, of size i)x2. Here f « 3 and g ■ 0. ile obtain by the construction in Theorem 3«5 -31 -31 31-: ence, suz'mn” i c i nr, w*. hf ’/e the f e 1 • cv;l nr result* Theorem 3.9# There exists a type II solution to the incidence equation for the finite protective plane case orders n » 10 and « 55 IV. Appendix Some Results on the Integral Solutions to the Incidence Equation for General v, h, X with gcd(k, k-X)» 1 and k-X Even. We consider here integral solutions to the incidence equation for general v, k, X, 0 < X < k < v - 1, under the conditions that gcd(k, k-X) ■ gcd(k, X) - 1 and k-X is even. Since (v-1) X » k(k-l) the first condition implies that JC _ v-1 k-1 k " X is an integer. This means that we can write ? k ■ ocX + 1 , v ■ c< X + oC + 1 . Since gcd(k,X) ■ 1 and k-X is even, both k and X must be odd. Also k-X ■ ( ec -1)X + 1 > 1 where ( oc -1)X is odd. This nears that oc > 2 and oC is even. We let A be such an integral solution which, without loss of generality, can be considered in canonical form where its column sums are nonnegative and the columns are in order by de creeing value of the column sums. Now (l‘.l) AAT - B =E(t) , and by Theorem. 3.3 in part II, T 0 .2) A A - B(w) Z [t{J] , where w « (Wj, ..., wv ) is such that (1.3) f A ■ k w , 9y (1,.2) -- ■ 5 6 (h.ii) t - X w.Wj+ (k-X) S Ly 1 < i* j < v, where £, Is the Kronecker delta. We let s^, 1 < i < v, denote the column i sum of A. Then by (k.3) and the canonical form of A - ' (b.5) s. » w.k, 1 < i < v, where w, > w_ > •.• v >0. Now by (L.L) and (L.5) 1 - 2 — v “ s? U-.6) t ^ m X + k - X, 1 < i < V. k Since pcd(k,X) ■ 1 and t^. Is an integer, (h.6) requires that 2 2 1 k divide s. whence k divide s.. This plus the canonical i i fcrm of A imply that all the w. are nonneaative integers. Now A may be a normal solution to the incidence equation, in which case all the w^'s are equal to 1. This means that A is a type I solution which is equivalent to a design matrix. From here on we consider only the nonnormal or type II solutions. Since U *7) ^ w2 ■ v , i-1 this means that some w. > 1 and some w . ■ 0. We then have the i J following result. Lemma L.l. If w. > 1 then ocV\+l > w. > oc . Proof: Let > 1. Since the entries in A are integers, 2 w. k « s. < t « X w + k - X . i i - ii i ' or 2 C1;.8) X w . - kw. + k - X>0. l l - 57 2 The two roots of the equation Xx-kx + k- X - 0 are x^ ■ oC - 1 + ^ and ' x » 1 . 2 If x^ > x > then Xx - kx + k- \<0. Since w .> 1 ■ x^, we rust then have by (U.8) Wj- Xx ■*- 1 *i • and since w^ is integral, Now by (h.*7) 2 2 . w . < v • oc X+oc+1, J or w . < oc-/x + 5 + — o < ocVx+i . We know that any column sum s of A must be represented by a sum of at most v nonzero integers. Let d^, dv be v variables which take integral values. We write them in a row D of v locations D - [d^, d^] , with the restriction ti&t (i:-9> £ *; - . . i-1 1 We let v 2 1 ■ z: w • i-l We define a T,LJ transforration------on the entries of D which does the following: d , d . + 1 and d . -* d - 1 11 J J where i j- j. Clearly this transformation preserves the sum s but it changes t by (L.10) - (d.+ l)2- d2+ (dj- l)2- d2 - 2(d.- dJ+ 1) . The following lemma determines the set of values for the entries in D which minimizes t for a fixed s. Lemma L.2. The value of t is a minimum if and only if (a) for s = 0 (mod v), each d^ jrt D has the same value s/v. (b) for s £ 0 (mod v), each dj in D has one of the two values d, d+1 where d _i_s the greatest integer less than s/v. Proof i Let s ~ 0 (mod v). If in D we have a d.< s/v, then by (L.9) we have a d. > s/v, and conversely. Here d^ + 1 < d. . The T. transformation yields by (L.10), J J j < 0 . Hence t is not a minimum on this D. But t > 0 is integral, hence it takes its minimum on some D which by elimination must be the one in (a). Let s ^ 0 (mod v). If in D we have a d^< d then by (L.9) we have a dj > d, and similarly i f a dj > d + 1 then a d^ < d + 1. By the same argument as before we can show that t is a minimum on the D in (b). Letting s ■ w k > 0, where w is an integer, we have the following lemma. Lemma 1:.3. We have (d+l)v>s>dv _if and only if (d + l)oC > w > dec . Proof: Assume (d+l)v>s>dv. This becomes 59 (d ♦ 1) v > w k > d v or Cb* 11) (d + l) ^ > v > d £ . Sir.cc v -oc k ♦ 1 and k « ocX ♦ 1, (1.11) becomes (1.12) w > d « + ^ > **** and (h.13) (d + l)oc + > w . k — Now by Leitma h.l, w < ofV^+l * hence by (h.12) d d+1 m _ ---- d+1 ^ < ------A +- T---- + 1 < 1 k oCX+1 oeX+1 whence from (lj.13) (d+l)o< ♦ 1 > w , and since w is an integer, (d+l)oc > w. Now assume (d+l)©c > w > dec . Then (d+l)ec + > w , k “ or (d+1) v > s . Since w < of-^/x+1 and etf > 2, X > 1, we have d <*/X+l <«X+1 whence d - i < i . oc X+l h Since w and doc are integers, w > dec + — ■ d — , k k whence s > d v , We can now prove the following general result. Theorem U.Jj. Either w^ ■ 1 or w. ~ 0 (modoc), 1 < i < v. A necessary condition that w^ ■ c*< j_s that a. no / i — I— i c < a/X + ■■■■ + . ■■■. p • — . * - v oc-1 2(*f-l) Proof t In a type II solution there is a w. *0 ~ 0 (modes) So let us consider only Wj > 0. Let (d+l)of > w. > dec, d > 0. Then by Lemma Iu3, (d+1) v > > d v, By Lemma L.2, the minimum value of t .. for such an s. is ii l (sj-dv)(d+l)2+ (v-Sj+dv)d2 2 3 2 2 2 3 • s.d +2s.d+s.- d v - 2d v - dv + vd - s,d + d v i l l i - d[2si- v(d+l)] + s., whence a . 15) t.j - Sj > d [2Sj- v(d+l) ] . Now by expressing t^ and Sj in terms of w. and, by Lemma It - 3, converting back to (d+l)oc > w. > d«c , (h.15) becomes ; ™ 1 Xwj-kwj + k - X > d [2kWj- v(d+l)] or 2 (L.16) f(wj) = X Wj - k(2d+l) Wj + k-X + vd(d+l) > 0 • Now letting Wj - d*t + 0 , 0 < fi < oC » k - * X + 1, and v • ocX+oc+1 in (lul6) we obtain (L.17) g(p) = \fi2- (2d+ocX+l)^9 - ofd2+ d2+ *cX-\ + d+l>0. We now examine the values of x for which g(x) > 0 is satisfied There must be some w. which, by Lemma L.l, must satisfy J d * < w, < #<^\+l, hence d <^X+1 or d2 < X, hence for x ■ 0, J 61 p(0) ■ - ocd2+ d2 + ocX-X + d+l « (X - d2).(oc- l) + d +. 1 > 0 . Fcr x • 1, 2 2 p (1) »\-2d-o<\ - 1-ocd + d + ocX-X + d+l • - d - d2(of- l), hence ** p(l) - C if d - 0, a(l) <0 if d > 0. For x - oc - 1, rr(oc -l) - X«x^- 2Xoe + X - 2 oc d + 2d -of2X +ocX -oc + 1 2 2 -ofd+d+ofX-X+d + 1 - -pc(d+l)2+ (d+1) (d+2) • -(d+1) [ (®f -2) + d(oc-l)], hence g(of-l) - a(l) * 0 i f of * 2 and d ■ 0. o(oC-l) < 0 if of > 2 cr d > 0 . For x * of , a(of) - Xoc2- 2dof - of2X -oc-o Ccnsiderino g(of) as a quadratic in d, we have u(of) > 0 if and only if d < -1 ♦ V1 *Of^T A * n~T>7 l:(«-lF "- 2 (oc 1 -1) Since w, » (d+l)of » c o< here, this bee ores g (°0 > 0 if 62 anti only it < v m r r i 0 .110 - v F i ra I ly, it i s cl car f r ona. i7) that for sufficiently larqe x > of » v c have o(x ) > 0. ’iow we have p(0)>0 , a(l) rernq, ti.ac far 1 < x g(yS) > 0 c n U for y8 ■ 1, a » C, and j S ■ Of • vvier the c cr.d i t i on (1 ♦ li ). ivine a f (i;?) is satir f ied only for w. » 1 or for i >0 — i \i | ■ cof vlere (i.li.) holds, uhc ncc th, rhco r;n. C, rol lary I , ' 6 * If r tl c rvr .h-.r of w-'s ouoo 1 ho 1, 2 then r ■ qof + of + 1 -a ere 0 < q < a . Proof: By The cron L.i, any w- ^ 1 ru^t be w.» v j°< , v- on inteoer. Then b . v , 2 _v-tL ? o v ■ of X-tof+1 « y ^ w - N ' v . of *" + r , i-1 i-i or v-r i "v 2 2 v x - y vf ) of + of + 1 ■ q of + of + 1 i = l ’ o r - (•*+!?. 1 Ci ' o<^ 'iv>. ir r - O then q in an intcoer while £_- ( « + l ) of <■’ act, H -21 ic >: r > 0 a nr’ ~ ince of > 1, q > 0. Since we are j;-s idcrina a type II solrt i on, q < X, 63 We also have the following result. Lenma L.6. Let yc be the number of w^'s such that w. « c«c and e be the greatest integer satisfying (L.lli). Then (L. 18) 2 L— yc c < x . c-l Frocf: By (ii.7) and Corollary L.5 e „ 2 2 2 / , yc c « < o< x , _ _ yc c c-l whence (L.16). With the aid of the foregoing results we. can construct sore examples of type II solutions for small values of v. For v * l $ t h - 7, X - 3 we can obtain three different type II solutions. Since c< - 2, c A - 1 1 .< 1 + 1 4 . The number of W £*S suer that w . - o< is then i 3. For 1, r - 11, we have 1 - b * 1L, t si ■ 11 * 16i ^ ♦ m 7, t.. < 12, 1 ■ 7- M j - .3, 2 < j , i 4 J; t-1 0, t.j o, 13 < i en si * " l ’ t ij ' > j A 1 » i 4 j; ttj - m < 1$\ 6’ ■ *■> ■ 0 , 2 f J f 12, 13 < and 61* 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 1 1 1 ■ ■ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 -1 r or 2, r - 7, we have 1 ■ ■ s t CM CM - 12; S2 ' li" 11 16‘ ‘l2 7, t.. - 7, t 3 < U j < 9, i / ji i j - 3’ O -*-> ■ h, t 10 < i, j < 15, i / j; ij " °' « 6, t . ■ t . m Of 1 < i < 2 , 3 < j < 9, 10 ' im J" ard 1 1 1 2 1 1 1 1 l l 1 l 1 i l l 1 1 l l l l l l l -1 l l l l -1 -l l l l 1 l -l l l l l l 1 l l l l l i 1 -l i l l 1 l -l -1 l l l l -1 1 l l- 1 l l 1 1 l 1 1 l l -1 l l l l l -1 -l l l l i -l l 65 For y » 3* r ■ 3* we have s. « lii, t.. - 16, - 12, 1 < i, j < 3, i 4 J> si " 7» tii • 7, t.j -• 3, U < i, j < 6, i ^ j; sd - 0, tH - L, tjj » 0, 7 < i, j < 15, i ^ j; tim “ 1 im “ °* 1 < 1 - 3j J < ? < m < 15; and 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 A 1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 I 1 -1 -1 1 1 -1 -1 -1 We also construct a type II solution for v * 23, k • 11, 1 p 5i 5i nee P( ■ 2, c < v T T T T T - 1 - 2, or c = 1, 2. Then y + ^ ’2 — "** 7'^ie P0S£ikle combinations are 1 < < 5 , = 0, and ^ “ 1* We construct a type II solution for this latter combination. For y^» y£ - 1, r - 3, we have S1 * '"k» tll “ s2 “ 22* fc22 " 26> tl? “ ^°; s. - 11, t.. « 11, tjj - 5, 3 < i, j < 5, i ^ j; 66 i ■ 0, tjj ■ 6, ■ 0, 6 < i, j < 23> i ^ jj t1£ - 20, t2i - 10, < i < 5, 6 < j < 23; and 1 2 1 1 1 1 2 ■ 1 -1 -1 2 1 -1 1 -1 2 1 -1 1 -1 2 ■ 1 1 -1 1 2 2 -1 1 1 1 -1 ■ 1 -1 -1 2 1-1 -11 2 1 - 1 - 1 1 A - 2 1 1 1 -1 2 2 -1 1 1 1 2 -1 1-1-1- 1 2 1 -1 - 1 1-1 2 1 -1 -1 1- 1 2 ■1 1 - 1-1 1 2 2 1 2 1 1 1 -1 1 2 -1 -1 -1 -1 1 o 1 -1 -1 1 -1 2 1 -1 -1 1 -1 2 1 ■ 1 1 1 -1 -1 V* References [I] Albert, A. A., "Rational Normal Matrices Satisfying the Incidence Equation," Proc. Amer. Math. Soc., U (1953), 551i-9. [2J Brauer, A., "On a New Class of Hadamard Determinants," Math. Zeit., 58 (1953), 219-2$. [3] Bruck, R. H. and H. J. Ryser, "The Nonexistence of Certain Finite Projective Planes," Can. Jour. Math., 1 (I9li9), 88-93. [Ji] Chowla, S, and H. J. RySer, "Combinatorial Problems," Can. Jour. Math., 2 (1950), 93-9. [5] Gantmacher, F. R., Matrix Theory, Vol. I, Chelsea, N.Y.(1959). [6] Hadamard, Jacques, "Resolution d'une Question Relative aux Determinants," Bull. d. Sciences Mathematiques (2), _17 (1893) part 1, 2ii0-6. [7] Hall, Marshall and H. J. Ryser, "Normal Completions of Incidence Matrices," Amer. Jour. Math., 76 (19510, 581-9. [8] Paley, R.E.A.C., "On Orthogonal Matrices," Jour. Math.'and Phys., 12 (1933), 311-20. [9] Ryser, H. J., "A Note on a Combinatorial Problem," Proc. Amer. Math. Soc., ^ (1950), L22-14. [10 ] ------, "Matrices with Integer Elements in Combinatorial Investigations," Amer. Jour. Math., 7h (1952), 769-73. [II] Scarpis, U*, "Sui Determinanti di Valore Massimo," Rend. d. R. Istituto Lombardo di Scienze e Lettere (2), H (1898), lLhl-6. [1?] Sylvester, J., "Thoughts on Inverse Orthogonal Matrices, simultaneous sign-Successions, and Tessellated Pavements in two or more colours, with applications to Newton's Rule, Ornamental Tile-work and the Theory of Numbers," Phil. Mag, (I), % (1867), L61-75. [13] Todd, J., "A Combinatorial Problem," Jour, Math, and Phys., 12 (1933), 321-33. [1L] Williamson, John, "Hadamard1s Determinant Theorem and the Sum of Four Squares," Duke Math. Jour., j_l (19U1*), 65-81. f15] ------"Note on Hadamard's Determinant Theorem," Bull. Amer. Math. Soc., 53 (191*7), 608-13. 68 AUTOBIOGRAPHY I, Eugene Carlyle Johnsen, was born in Minneapolis, Minnesota, January 27, 1932. There I received my secondary school education at the Theodore Roosevelt Senior High School, and my undergraduate training in chemistry at the University of Minnesota, vihich granted me the degree Bachelor of Chemistry in 19$b* For three years I was a graduate student in mathematics and a teaching assistant and instructor of chemistry and mathematics in the General College of that university. In 1957 I entered the Graduate School of The Chio State University in the Department of Mathematics and taught under graduate mathematics for one year. From 1958 through 1961 I was a research assistant in algebra under Professor K. J. Ryser, and during the summer of 1959 held a National Science Foundation r'ellowship, while comrleting the requirements for the degree Doctor of Philosophy.