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Discrete Mathematics 97 (1991) 387-393 387 North-Holland
Designs constructed from maximal arcs
D.R. Stinson* Department of Computer Science, University of Manitoba, Winnipeg, Man., Canada R3T 2N2 Current address: Computer Science and Engineering, University of Nebraska, Lincoln, NE 68588, USA
Received 7 March 1990 In memory of Egmont Kiihler.
Abstract Stinson, D.R., Designs constructed from maximal arcs, Discrete Mathematics 97 (1991) 387-393. In this paper, we give constructions for various types of designs using maximal arcs in projective planes. In particular, we construct resolvable BIBDs and group-divisible designs having orthogonal resolutions. We also define a new type of resolution of designs that we call a hyperresolution. Examples are obtained from the maximal arcs of Denniston.
1. Introduction
A (v, k, l)-balanced incomplete block design (or (v, k, l)-BIBD), is a pair (X, a), where & is a set of k-subsets (called blocks) of the v-set X, such that every pair of points occurs in a unique block. A projective plane of order q is a (q* + q + 1, q + 1, l)-BIBD. The blocks of a projective plane are usually called lines. It is well known that a projective plane of order q exists if q is a prime power. A maximal {w, n}-arc in a projective plane of order q is a set of w points in the plane such that every line intersects the maximal arc in n points or in 0 points. It follows that w = 1 + (n - l)(q + 1) in a maximal arc. Denniston has proved the existence of the following infinite class of maximal arcs in Desarguesian projective planes of even order.
Theorem 1.1 [8]. For all 0 In the special case s = r + 1, the arcs were previously constructed in [6]. We also note that Thas has shown in [17] that there is no (2q + 3, 3}-arc in PG(2, q) whenq=3handh>1. * This research was supported by NSERC grant A9287. 0012-365X/91/$03.50 @ 1991 -Elsevier Science Publishers B.V. All rights reserved 388 D. R. Stinson In this paper we discuss several interesting combinatorial designs that can be constructed from maximal arcs. 2. Resolvable designs A (v, k, l)-BIBD is defined to be resolvable if the set of blocks can be partitioned into parallel classes, each of which consists of v/k disjoint blocks. Clearly, a maximal {w, n}-arc in a projective plane of order 4 gives rise to a (w, IZ, l)-BIBD (see, for example, [14]). In fact, Wallis proved in [19] that this BIBD is resolvable, as follows. Theorem 2.1 [19]. If there exists a maximal {w, n}-arc in a projective plane of order q, then there is a resolvable (w, n, l)-BIBD. Proof. Let (X, &) be a projective plane of order q and let Y G X be the points in a maximal {w, n}-arc. Then (Y, {A fl Y: A E a}) is a (w, 12, l)-BIBD. For any point x E X\Y, define P, = {A fl Y: x E A E &}. Then, P, is a parallel class. Let L be some line that is disjoint from Y. Define 9(L) = {Px: x E L}. It is easy to verify that 9(L) is a resolution of the BIBD. q Corollary 2.2. For all 0 < r s s, there is a resolvable (2’+’ + 2’ - 2”, 2’, l)-BIBD. Proof. Theorems 1.1 and 2.1. Cl When s = r + 1, we get the resolvable BIBDs constructed by Seiden [15] (the corresponding maximal arcs were first constructed in [6]). We now consider more general designs. A group-divisible design (or GDD), is a triple (X, %, &), which satisfies the following properties: (1) % is a partition of X into subsets called groups, (2) d is a set of subsets of X (called blocks) such that a group and a block contain at most one common point, and (3) every pair of points from distinct groups occurs in a unique block. The group-type (or type) of a GDD (X, 3, a) is the multiset {ICI: G E %}. The multiset consisting of one integer t occurring u times will be denoted tU. We will say that a GDD is a k-GDD if IAl = k for every A E ~4. Note that a (v, k, l)-BIBD is just a k-GDD of type 1”. As with BIBDs, we say that a k-GDD is resolvable if the set of blocks can be partitioned into parallel classes. A necessary condition for a k-GDD to be resolvable is that every group has the same size; hence the type will be tU for some t and u. It is also clearly necessary that k ) tu. Let $8 and Y be two resolutions of a BIBD or GDD. We say that these resolutions are orthogonal if, for any parallel classes R E 3 and S E Y, R and S Designs constructedfrom maximal arcs 389 contain at most one common block. Orthogonal resolutions have been studied in several papers; a good survey is given by Lamken and Vanstone [lo]. We have the following construction for GDDs with orthogonal resolutions. Theorem 2.3. Zf 0 < r s s, then there existi a 2’-GDD of type (2r)T-2’-‘+1 having 2”-’ orthogonal resolutions. Proof. Let (X, a) be a projective plane of order 2’ and let Y E X be the points in a maximal {w, n}-arc. Let x E X\Y; then (Y, {A~-IY;xEAE&}, {A~~Y:x$AE&}) is a 2’-GDD of type (T)r-2’-‘+1. For any line L such that L is disjoint from Y and x E L, define 9(L)={P,:y~L,y#x}, whereP,={AnY:yeAE&}. Then, any such S(L) is a resolution of the GDD, and S(L) is orthogonal to 2(L’) if L#L’(wherexEL, XEL’). Cl There has been some recent interest in the maximum number of orthogonal resolutions a GDD can have. The following general upper bound is proved in [W. Theorem 2.4 [lo]. Zf u > k, then the maximum number of orthogonal resolutions of a k-GDD of type t“ is at most t(u - l)/(k - 1) -k. In the case of the GDDs constructed in Theorem 2.3, we have obtained 2”-’ orthogonal resolutions, whereas the upper bound given by Theorem 2.4 is 2” - 2 orthogonal resolutions. Hence, there is still some room for improvement. 3. Hyperresolutions In this section, we define a new type of object we call a hyperresolution. A hyperresolution of a (v, k, l)-BIBD is a set of parallel classes 9 such that, for every two disjoint blocks A and B in the BIBD, there is a unique parallel class P E 9 such that {A, B} c P. This concept generalizes the hyperfactorizations studied by Jungnickel and Vanstone in [9]; a hyperfactorization of K2,, is the same thing as a hyperresolution of a (2n, 2, 1)BIBD. Maximal arcs in projective planes can be used to construct hyperresolutions, as follows. Theorem 3.1. Zf there exists a maximal {w, n}-arc in a projective plane of order q, then there is a hyperresolution of a (w, n, l)-BIBD. Proof. Let (X, ~4) be a projective plane of order q and let Y c X be the points in a maximal {w, n}-arc. As before (Y, {A fl Y: A E d}) is a (w, n, l)-BIBD, and for 390 D. R. Stinson any point x E X\Y, P, = {A fl Y: x E A E sd} is a parallel class. Define $Zrj= {Px: x E X\Y}. We shall verify that 8 is a hyperresolution of the BIBD. Let A n Y and A’ fl Y be disjoint blocks in the BIBD. Since (X, .J$) is a projective plane, A and A’ meet in a unique point x E X\Y. Then P, is the unique parallel class containing A II Y and A’ n Y. 0 Corollary 3.2. For all 0 < r s s, there is a (Tts + 2’ - 2”, 2’, l)-BIBD having a hyperresolution. When r = 1 in Corollary 3.2, we get a hyperfactorization of K,$+,[9]. We also remark that such a hyperfactorization of K, was used in [2] to construct PG(2,4) from one of its hyperovals. Hyperresolutions seem to be of interest in their own right, but they can also be used to construct partial geometries. A partial geometry pg(K, R, T) is a pair (X, a), which satisfies the following properties: (1) ti is a set of subsets of X (called lines), each of cardinality K, (2) every point occurs on exactly R lines, (3) every pair of points occur on at most one line, (4) if x is a point and L is a line such that x 4 L, then there are exactly T points y such that y E L and x and y are collinear. For information on partial geometries, we refer to [3,5,18,7]. Theorem 3.3. Zfthere is a hyperresolution of a (v, k, l)-BIBD, then there is a partial geometry v v-k2+k-1 v k-l ‘k- Proof. Let (X, a) be the BIBD and let 9 be the hyperresolution. We shall prove that (~4, 9) is a partial geometry. First, every parallel class has v/k blocks in it, so K = v/k. Next, every block A meets k(v - l)/(k - 1) other blocks, so it is disjoint from v(v - 1)_ 1 _ k(v - 1) k(k - 1) k-l blocks. (v - k)/k of these blocks occur in each parallel class in 9 containing A. Taking the quotient of these two quantities, we see that R=v-k2+k-l k-l ’ Finally, let P be a parallel class in 9’ and let A be a block not in P. A meets k of the v/k blocks in P. For each of the remaining v/k - k blocks A’ E P, there is a different parallel class containing A and A’. Hence, T = v/k - k. q Designs constructed from maximal arcs 391 Corollary 3.4 [16]. For all 0 < r s s, there is a partial geometry pg(2” - 2”_’ + 1, 2” - 2’ + 1, (2’ - 1)(2”_‘- 1)). Proof. Corollary 3.2 and Theorem 3.3. q We note that the construction of partial geometries from maximal arcs is due to Thas [16] and Wallis [19]. What we have done here is to observe that the construction can be broken into two steps, namely maximal arc+ hyperresolution + partial geometry. Corollary 3.4 provides hyperresolutions of BIBDs where the block size is a power of 2. It is easy to see that any (q*, q, l)-BIBD has a hyperresolution, for such a BIBD is an affine plane and the (unique) resolution of the BIBD is a hyperresolution. No other examples of hyperresolutions are known to the author. We can make a couple of observations on the first possible cases of block size 3. As noted above, a (9,3, l)-BIBD has a hyperresolution. The next case to consider is parameters (15,3,1). We have the following. Lemma 3.5. There exists no (15, 3, l)-BIBD having a hyperresolution. Proof. We have two proofs of this assertion. First, existence of a hyperresolution of a (15,3, l)-BIBD would imply the existence of a pg(5,4,2); but this geometry is known not to exist [5,21]. The second proof is to verify directly from the list of all 80 non-isomorphic (15,3, l)-BIBDs given in [12] that no hyperresolution exists. A hyperresolution would consist of 28 parallel classes, Of these 80 designs, only two have this many parallel classes: design SC1 has 56 parallel classes, and design #7 has 32 parallel classes. A minute’s reflection reveals that it is impossible to choose 28 parallel classes which form a hyperresolution from either of these two designs. 0 Remark. The 56 parallel classes of design SC1have the property that every pair of disjoint blocks occur in exactly 2 parallel classes. In a similar fashion as in [lo], we might define such an object to be a 2-hyperresolution. This allows us to construct a generalized partial geometry pg,(5,4,2) (for a definition, see [lo]). The next case is that of a (21,3, l)-BIBD. A hyperresolution of this design would imply the existence of a partial geometry pg(7,7,4). The existence of this partial geometry is listed in [5] as being unknown. 4. An example Let’s briefly discuss the designs obtained from the {28,4}-arc in a projective plane of order 8. From it, we get a resolvable (28,4, l)-BIBD, a 4-GDD of type 392 D.R. Stinson 47 having 2 orthogonal resolutions, and a (28,4, l)-BIBD having a hyperresolution. The (28,4,1)-design obtained from the Denniston arc is referred to as the ‘Ree Unital’ in Brouwer’s list of (28,4, l)-BIBDs [4]; see also [13]. Brouwer indicates that this BIBD has exactly 45 parallel classes, and precisely 10 resolutions. The 45 parallel classes are those in the hyperresolution, and the 10 resolutions are constructed using Theorem 2.1 from the 10 lines which are disjoint from the arc. Brouwer’s results tell us that there are no parallel classes or resolutions other than these. It is also mentioned in [4] that the automorphism group of this (28,4,1)-BIBD is PI’L(2,8) (having order 1512), and that this group is doubly transitive on the points of the design. By Theorem 2.3, we obtain 2 orthogonal resolutions of a 4-GDD of type 47. The results mentioned above tell us that it is impossible to find 3 orthogonal resolutions of a 4-GDD of type 47 using this particular (28,4,1)-design. Theorem 2.4 tells us that no 4-GDD of type 4’ can have 5 orthogonal resolutions. However, we can also rule out the possibility of 4 orthogonal resolutions as follows. Theorem 2.5. The maximum number of orthogonal resolutions of a 4-GDD of type 4’ is either 2 or 3. Proof. By [ll], The existence of a 4-GDD of type 47 having 4 orthogonal resolutions would imply the existence of an 8-GDD of type 415 (i.e. an elliptic semiplane S(60, 8, 4); see Baker [l]). However, Baker proves in [l] that this elliptic semiplane does not exist. Cl Acknowledgement I would like to thank the referee for supplying several useful references. Added in proof In [20], De Clerck presents some results similar to those proved here. In particular, he gives a construction similar to Theorem 3.3. References [l] R.D. Baker, Elliptic semi-planes I: existence and classification, Proc. 8th S-E Conf. Combin., Graph Theory and Comput., Baton Rouge (1977) 61-73. [2] A. Beutelspacher, 21- 6 = 15: a connection between two distinguished geometries, Amer. Math. Monthly 93 (1986) 29-41. Designs constructed from maximal arcs 393 [3] R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963) 389-419. [4] A.E. Brouwer, Some unitals on 28 points and their embeddings in projective planes of order 9, in: Geometries and Groups, Lecture Notes in Math., Vol. 893 (Springer, Berlin, 1981) 183-188 (a preliminary version appeared as Math. Centre report ZW155, Amsterdam, 1981). (51 A.E. Brouwer and J.H. van Lint, Strongly regular graphs and partial geometries, in: D.M. Jackson and S.A. Vanstone, eds., Enumeration and Design (Academic Press, New York, 1984) 85-122. [6] A. Cossu, Su alcune proprieta dei (k; n)-archi di un piano proiettivo su un corpo finite, Rend. Mat. 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