Self orthogonal Knut Vik designs

The purpose of this note is twofold. First we show that the transpose of a cyclic Knut Vik design is a Knut Vik design. Second we prove that all the cyclic Knut Vik designs are self orthogonal.     

Orthogonal designs II

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1,k) and ordern exist for everyk < n whenn = 2 ^t+2?3 andn = 2 ^t+2?5 (wheret is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that theweighing matrix conjecture ‘For every ordern ≡ 0 (mod 4) there is, for eachk ?n, a square {0, 1, ? 1} matrixW = W(n, k) satisfyingWW ^t =kIn’ is resolved in the affirmative for all ordersn = 2^t+1?3,n = 2^t+1?5 (t a positive integer). The fact that the matrices we find are skew-symmetric for allk < n whenn ≡ 0 (mod 8) and because of other considerations we pose three other conjectures about weighing matrices having additional structure and resolve these conjectures affirmatively in a few cases. In an appendix we give a table of the known results for orders ? 64.     

Group divisible designs with block size four and two groups

We give some constructions of new infinite families of group divisible designs, GDD(n,2,4;λ₁,λ₂), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3?n?8. For n=10 there is one missing critical design. If λ₁>λ₂, then the necessary conditions are sufficient for . For each of n=10,15,16,17,18,19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.     

Some designs which admit strong tactical decompositions

Whenever there exist affine planes of orders n ? 1 and n, a construction is given for a 2 ? ((n + 1)(n ? 1)², n(n ? 1), n) design admitting a strong tactical decomposition. These designs are neither symmetric nor strongly resolvable but can be embedded in symmetric 2 ? (n³ ? n + 1, n², n) designs.     

Large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ have been studied by Schellenberg, Chen, Lindner and Stinson. These large sets have applications in cryptography in the construction of perfect threshold schemes. It is known that such large sets can exist only for n ≡ 0 (mod 3) and do exist for n = 6 and for all n = 3^k, k ≥ 1. In this paper, we present new recursive constructions and use them to show that such large sets exist for all odd n ≡ 0 (mod 3) and for even n = 24m, where m odd ≥ 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 285–296, 2001     

Block designs and graph imbeddings

Heffter first observed that certain imbeddings of complete graphs give rise to BIBD's with k = 3 and λ = 2 (and sometimes λ = 1); Alpert established a one-to-one correspondence between BIBD's with k = 3 and λ = 2 and triangulation systems for complete graphs. In this paper we extend this correspondence to PBIBD's on two association classes with k = 3, λ₁ = 0 and λ₂ = 2, and triangulation systems for strongly regular graphs. The group divisible designs of Hanani are used to construct triangulations for the graphs K_n(m), in each case permitted by the euler formula. Conversely, triangular imbeddings of K_n(m) are constructed which lead to new group divisible designs. A process is developed for “doubling” a given PBIBD of an appropriate form. Various extensions of these ideas are discussed, as is an application to the construction of quasigroups.     

Asymptotic existence theorems for frames and group divisible designs

In this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs.     

Some remarks on permutation designs

A definition of isomorphism of two permutation designs is proposed, which differs from the definition in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392]. The proposed definition has the (generally required) property that the allowed permutations always transform a permutation design into a permutation design. It is shown that the n permutation designs coming from the partitioning of S_n into permutation designs, as constructed in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392] are all isomorphic. Further we find that this modified definition does not increase the number of nonisomorphic (6, 4) permutation designs. The same investigation showed that one of the designs, claimed to be a (6, 4) permutation design in [J. Combinatorial Theory Ser. A21 (1976), 384–392], is actually not a (6, 4) permutation design.     

Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

The existence of Howell designs of odd side

Three recursive constructions for Howell designs are described, and it is shown that all Howell designs H(s, 2n) exist, with s odd, n ? s ? 2n ? 1, (s, 2n) ≠ (3, 4), (5, 6), or (5, 8). This settles the existence question for Howell designs of odd side.     

Embeddable transversal designs

The embeddability of certain (group) divisible designs in symmetric 2-designs is investigated. These designs are symmetric resolvable transversal designs. It is proved that all such transversal designs with v = 2k are embeddable and some necessary and sufficient conditions for other cases are given.     

On resolvable designs S3(3; 4, v)

We study a method of Lonz and Vanstone which constructs an S₃(3, 4, 2n) from any given 1-factorization of K_2n. We show that the resulting designs admit at least 3 mutually orthogonal resolutions whenever n ⩾ 4 is even. In particular, the necessary conditions for the existence of a resolvable S₃(3, 4, ν) are also sufficient. Examples without repeated blocks are shown to exist provided that n ≢ 2 mod 3.     

D-optimal designs embedded in Hadamard matrices and their effect on the pivot patterns

In this paper we develop a new approach for detecting if specific D-optimal designs exist embedded in Sylvester-Hadamard matrices. Specifically, we investigate the existence of the D-optimal designs of orders 5, 6, 7 and 8. The problem is motivated to explaining why specific values appear as pivot elements when Gaussian elimination with complete pivoting is applied to Hadamard matrices. Using this method and a complete search algorithm we explain, for the first time, the appearance of concrete pivot values for equivalence classes of Hadamard matrices of orders n = 12, 16 and 20.     

The application of invariant theory to the existence of quasi-symmetric designs

Gleason and Mallows and Sloane characterized the weight enumerators of maximal self-orthogonal codes with all weights divisible by 4. We apply these results to obtain a new necessary condition for the existence of 2 − (v, k, λ) designs where the intersection numbers s₁…,s_n satisfy s₁ ≡ s₂ ≡ … ≡ s_n (mod 2). Non-existence of quasi-symmetric 2−(21, 18, 14), 2−(21, 9, 12), and 2−(35, 7, 3) designs follows directly from the theorem. We also eliminate quasi-symmetric 2−(33, 9, 6) designs. We prove that the blocks of quasi-symmetric 2−(19, 9, 16), 2−(20, 10, 18), 2-(20,8, 14), and 2−(22, 8, 12) designs are obtained from octads and dodecads in the [24, 12] Golay code. Finally we eliminate quasi-symmetric 2−(19,9, 16) and 2-(22, 8, 12) designs.     

Two-dimensional minimax Latin hypercube designs

We investigate minimax Latin hypercube designs in two dimensions for several distance measures. For the ?^∞-distance we are able to construct minimax Latin hypercube designs of n points, and to determine the minimal covering radius, for all n. For the ?¹-distance we have a lower bound for the covering radius, and a construction of minimax Latin hypercube designs for (infinitely) many values of n. We conjecture that the obtained lower bound is attained, except for a few small (known) values of n. For the ?²-distance we have generated minimax solutions up to n=27 by an exhaustive search method. The latter Latin hypercube designs are included in the website www.spacefillingdesigns.nl.     

New Z-cyclic triplewhist frames and triplewhist tournament designs

Triplewhist tournaments are a specialization of whist tournament designs. The spectrum for triplewhist tournaments on v players is nearly complete. It is now known that triplewhist designs do not exist for v=5,9,12,13 and do exist for all other except, possibly, v=17. Much less is known concerning the existence of Z-cyclic triplewhist tournaments. Indeed, there are many open questions related to the existence of Z-cyclic whist designs. A (triple)whist design is said to be Z-cyclic if the players are elements in Z_m∪A where m=v, A=∅ when and m=v-1, A={∞} when and it is further required that the rounds also be cyclic in the sense that the rounds can be labelled, say, R₁,R₂,… in such a way that R_j+1 is obtained by adding to every element in R_j. The production of Z-cyclic triplewhist designs is particularly challenging when m is divisible by any of 5,9,11,13,17. Here we introduce several new triplewhist frames and use them to construct new infinite families of triplewhist designs, many for the case of m being divisible by at least one of 5,9,11,13,17.     

On construction of partially balanced n-ary designs

A new method of constructing a series of partially balanced ternary (PBT) designs is presented. In the method, we have added the corresponding rows of incidence matrices of a BIB design and a PBIB design, both obtained from single initial block with at least one element in common between them. The BIB and PBIB designs above were obtained by method of differences. We have also constructed PBT designs and PB n-ary designs from a PBIB design alone based on NC_m-scheme as well as from a group divisible PBIB design with smaller number of blocks and moderate block sizes.     

Counting and constructing orthogonal circulants

If F is an arbitrary finite field and T is an n × n orthogonal matrix with entries in F then one may ask how to find all the orthogonal matrices belonging to the algebra F[T] and one may want to know the cardinality of this group. We present here a means of constructing this group of orthogonal matrices given the complete factorization of the minimal polynomial of T over F. As a corollary of this construction scheme we give an explicit formula for the number of n × n orthogonal circulant matrices over GF(p^l) and a similar formula for symmetric circulants. These generalize results of MacWilliams, J. Combinatorial Theory10 (1971), 1–17.     

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

A weak difference set construction for higher dimensional designs

We continue the analysis of de Launey's modification of development of designs modulo a finite groupH by the action of an abelian extension function (AEF), and of the proper higher dimensional designs which result.We extend the characterization of allAEFs from the cyclic group case to the case whereH is an arbitrary finite abelian group.We prove that ourn-dimensional designs have the form (f(j ₁ j ₂ ...j _n )) (j _i J), whereJ is a subset of cardinality |H| of an extension groupE ofH. We say these designs have a weak difference set construction.We show that two well-known constructions for orthogonal designs fit this development scheme and hence exhibit families of such Hadamard matrices, weighing matrices and orthogonal designs of orderv for which |E|=2v. In particular, we construct proper higher dimensional Hadamard matrices for all orders 4t100, and conference matrices of orderq+1 whereq is an odd prime power. We conjecture that such Hadamard matrices exist for all ordersv0 mod 4.