Showing papers in "Journal of Combinatorial Designs in 2005"
TL;DR: Four Turyn type sequences of lengths 36, 36,36, 35, 35 are found by a computer search and used to generate a number of new T‐sequences, creating many new Hadamard matrices constructible using these new T-sequences.
Abstract: Four Turyn type sequences of lengths 36, 36, 36, 35 are found by a computer search. These sequences give new base sequences of lengths 71, 71, 36, 36 and are used to generate a number of new T-sequences. The first order of many new Hadamard matrices constructible using these new T-sequences is 428. © 2004 Wiley Periodicals, Inc.
160 citations
TL;DR: In this article, the authors consider the class of I-graphs I(n,j,k), which is a generalization over the generalized Petersen graphs, and study different properties of Igraphs, such as connectedness, girth, and whether they are bipartite or vertex-transitive.
Abstract: We consider the class of I-graphs I(n,j,k), which is a generalization over the class of the generalized Petersen graphs. We study different properties of I-graphs, such as connectedness, girth, and whether they are bipartite or vertex-transitive. We give an efficient test for isomorphism of I-graphs and characterize the automorphism groups of I-graphs. Regular bipartite graphs with girth at least 6 can be considered as Levi graphs of some symmetric combinatorial configurations. We consider configurations that arise from bipartite I-graphs. Some of them can be realized in the plane as cyclic astral configurations, i.e., as geometric configurations with maximal isometric symmetry. © 2005 Wiley Periodicals, Inc.
63 citations
TL;DR: The best known upper bound for covering arrays is 2-CA (n, k, g) as discussed by the authors, where g is a set of g symbols with the property that in each subarray, every column appears at least once.
Abstract: A covering array t-CA (n, k, g) is a k × n array on a set of g symbols with the property that in each t × n subarray, every t × 1 column appears at least once. This paper improves many of the best known upper bounds on n for covering arrays, 2-CA (n, k, g) with g + 1 ≤ k ≤ 2g, for g = 3 · · · 12 by a construction which in many of these cases produces a 2-CA (n, k, g) with n = k (g − 1) + 1. The construction is an extension of an algebraic method used by Chateauneuf, Colbourn, and Kreher which uses an array and a group action on the array. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 70–77, 2005.
59 citations
TL;DR: In this article, the existence of 4-RGDDs and uniform 5-GDDs was investigated and it was proved that the necessary conditions for such designs are also sufficient with a finite number of possible exceptions.
Abstract: In this paper, we continue to investigate the existence of 4-RGDDs and uniform 5-GDDs. It is proved that the necessary conditions for the existence of such designs are also sufficient with a finite number of possible exceptions. As an application, the known results on the existence of uniform 4-frames are also improved. © 2004 Wiley Periodicals, Inc.
54 citations
TL;DR: Several direct constructions via skew starters and Weil's theorem on character sum estimates are given in this paper, improving the known existence results on optimal OOCs.
Abstract: Several direct constructions via skew starters and Weil's theorem on character sum estimates are given in this paper for optimal (gv, 5, 1) optical orthogonal codes (OOCs) where 60 ≤ g ≤ 180 satisfying g ≡ 0 (mod 20) and v is a product of primes greater than 5. These improve the known existence results on optimal OOCs. Especially, we provide an optimal (v, 5, 1)-OOC for any integer v ≡ 60, 420, 660, 780, 1020, 1140, 1380, 1740 (mod 1800). © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 54–69, 2005.
47 citations
TL;DR: In this paper, the authors studied the relationship between the symmetries of the starting factorization and the automorphisms of the end factorization, and showed that the autotopies inherited by the K-construction can be inherited as autotypes by the end product, either as automorphism of the factorization or autotopy of the latin square.
Abstract: A 1-factorization of a graph is a decomposition of the graph into edge disjoint perfect matchings. There is a well-known method, which we call the K-construction, for building a 1-factorization of Kn;n from a 1-factorization of Knþ1. The 1-factorization of Kn;n can be written as a latin square of order n. The K-construction has been used, among other things, to make perfect 1-factorizations, subsquare-free latin squares, and atomic latin squares. This paper studies the relationship between the factorizations involved in the K-construction. In particular, we show how symmetries (automorphisms) of the starting factorization are inherited as symmetries by the end product, either as automorphisms of the factorization or as autotopies of the latin square. Suppose that the K-construction produces a latin square L from a 1-factorization F of Knþ1. We show that the main class of L determines the isomorphism class of F, although the converse is false. We also prove a number of restrictions on the symmetries (autotopies and paratopies) which L may possess, many of which are simple consequences of the fact that L must be symmetric (in the usual matrix sense) and idempotent. In some circumstances, these restrictions are tight enough to ensure that L has trivial autotopy group. Finally, we give a cubic time algorithm for deciding whether a main class of latin squares contains any square derived from the K-construction. The algorithm also detects symmetric squares and totally symmetric squares (latin squares that equal their six conjugates). # 2005 Wiley Periodicals, Inc. J Combin Designs 13: 157-172, 2005.
30 citations
TL;DR: In this article, a generalization of Buekenhout's construction of unitals is presented, which gives minimal blocking sets of size q4/3/3 + 1 or q 4/3+1 + 2 in non-prime order.
Abstract: The size of large minimal blocking sets is bounded by the Bruen–Thas upper bound. The bound is sharp when q is a square. Here the bound is improved if q is a non-square. On the other hand, we present some constructions of reasonably large minimal blocking sets in planes of non-prime order. The construction can be regarded as a generalization of Buekenhout's construction of unitals. For example, if q is a cube, then our construction gives minimal blocking sets of size q4/3 + 1 or q4/3 + 2. Density results for the spectrum of minimal blocking sets in Galois planes of non-prime order is also presented. The most attractive case is when q is a square, where we show that there is a minimal blocking set for any size from the interval . © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 25–41, 2005.
25 citations
TL;DR: In this article, Combin et al. generalized the concept of cyclic balanced sampling plans to a cyclic BSA (ν, 3, λ; α) with α = 2, 3.
Abstract: Balanced sampling plans excluding contiguous units (or BSEC) were first introduced by Hedayat, Rao, and Stufken in 1988. In this paper, we generalize the concept of a cyclic BSEC to a cyclic balanced sampling plan to avoid the selection of adjacent units (or CBSA for short) and use Langford and extended Langford sequences to construct a cyclic BSA(ν, 3, λ; α) with α = 2, 3. We finally establish the necessary and sufficient conditions for the existence of a cyclic BSA(ν, 3, λ; α) where α = 2, 3. © 2005 Wiley Periodicals, Inc. J Combin Designs.
19 citations
TL;DR: In this article, Combin et al. gave a positive answer for new classes of groups, namely nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian sylow 2 -subgroup which possesses a cyclic subgroup of index 2.
Abstract: For which groups G of even order 2n does a 1-factorization of the complete graph K2n exist with the property of admitting G as a sharply vertex-transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in 4, we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian Sylow 2-subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1-factor. © 2005 Wiley Periodicals, Inc. J Combin Designs
17 citations
TL;DR: A backtracking over parallel classes with a partial isomorph rejection (PIR) is carried out to enumerate the resolvable 2‐(10,5,16) designs, showing that the inclusion of PIR reduce substantially the CPU time for the enumeration of all designs.
Abstract: A backtracking over parallel classes with a partial isomorph rejection (PIR) is carried out to enumerate the resolvable 2-(10,5,16) designs. Computational results show that the inclusion of PIR reduce substantially the CPU time for the enumeration of all designs. We prove first some results, which enable us to restrict the search space. Since every resolvable 2-(10,5,16) design is also a resolvable 3-(10,5,6) design and vice versa, the latter designs are also enumerated. There are 27, 121, 734 such designs with automorphism groups whose order range from 1 to 1,440. From these, designs 2,006,690 are simple. © 2004 Wiley Periodicals, Inc.
15 citations
TL;DR: A new array of order 12 is introduced, which is suitable for any set of four amicable circulant matrices, and applied to construct new self‐dual codes over large finite fields, which lead to the odd Leech lattice by Construction A.
Abstract: Symmetric designs and Hadamard matrices are used to construct binary and ternary self-dual codes. Orthogonal designs are shown to be useful in construction of self-dual codes over large fields. In this paper, we first introduce a new array of order 12, which is suitable for any set of four amicable circulant matrices. We apply some orthogonal designs of order 12 to construct new self-dual codes over large finite fields, which lead us to the odd Leech lattice by Construction A. © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 184–194, 2005.
TL;DR: This paper constructs superimposed codes from super‐simple designs which give results better than superimposed code constructed by other known methods and improves numerical values of upper bounds for the asymptotic rate of some (w,r) superimposed Codes.
Abstract: A (w,r) cover-free family is a family of subsets of a finite set such that no intersection of w members of the family is covered by a union of r others. A (w,r) superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as the concept of key distribution pattern. In the present paper, we give some new results on superimposed codes. First we construct superimposed codes from super-simple designs which give us results better than superimposed codes constructed by other known methods. Next we prove the uniqueness of the (1,2) superimposed code of size 9 × 12, the (2,2) superimposed code of size 14 × 8, and the (2,3) superimposed code of size 30 × 10. Finally, we improve numerical values of upper bounds for the asymptotic rate of some (w,r) superimposed codes. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, the existence of (v, K)-pairwise balanced designs (PBDs) for a few sets K of prime powers ≥ 8 and also for a number of subsets K of {5, 6, 7, 8, 9}, which contain {5}.
Abstract: In this paper, we look at the existence of (vK) pairwise balanced designs (PBDs) for a few sets K of prime powers ≥ 8 and also for a number of subsets K of {5, 6, 7, 8, 9}, which contain {5}. For K = {5, 7}, {5, 8}, {5, 7, 9}, we reduce the largest v for which a (v, K)-PBD is unknown to 639, 812, and 179, respectively. When K is Q≥8, the set of all prime powers ≥ 8, we find several new designs for 1,180 ≤ v ≤ 1,270, and reduce the largest unsolved case to 1,802. For K =Q0,1,5(8), the set of prime powers ≥ 8 and ≡ 0, 1, or 5 (mod 8) we reduce the largest unknown case from 8,108 to 2,612. We also obtain slight improvements when K is one of {8, 9} or Q0,1(8), the set of prime powers ≡ 0 or 1 (mod 8). © 2004 Wiley Periodicals, Inc.
TL;DR: In this paper, a recursive construction for anti‐mitre Steiner triple systems is presented and another construction of anti-mitre STSs by utilizing 5‐sparse ones is presented.
Abstract: In this paper, we present a recursive construction for anti-mitre Steiner triple systems. Furthermore, we present another construction of anti-mitre STSs by utilizing 5-sparse ones. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, the existence of disjoint group-divisible designs with block size three and type 2n41 (denoted by LS (2n41)) was shown.
Abstract: Large sets of disjoint group-divisible designs with block size three and type 2n41 (denoted by LS (2n41)) were first studied by Schellenberg and Stinson and motivated by 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 any n ∉ {12, 36, 48, 144} ∪ {m > 6 : m ≡ 6,30 (mod 36)}. In this paper, we show that an LS (212k + 641) exists for any k ≠ 2. So, the existence of LS (2n41) is almost solved with five possible exceptions n ∈ {12, 30, 36, 48, 144}. This solution is based on the known existence results of S (3, 4, v)s by Hanani and special S (3, {4, 6}, 6m)s by Mills. Partitionable H (q, 2, 3, 3) frames also play an important role together with a special known LS (21841) with a subdesign LS (2641). © 2004 Wiley Periodicals, Inc.
TL;DR: The notion of a generalized candelabra t‐system is introduced and used to show that a T*(3, 4, v)‐code exists for all odd v and, combined with Levenshtein's result, the existence problem for a T-code is solved completely.
Abstract: A word of length k over an alphabet Q of size v is a vector of length k with coordinates taken from Q. Let Q*4 be the set of all words of length 4 over Q. A T*(3, 4, v)-code over Q is a subset C*⊆ Q*4 such that every word of length 3 over Q occurs as a subword in exactly one word of C*. Levenshtein has proved that a T*(3, 4, vv)-code exists for all even v. In this paper, the notion of a generalized candelabra t-system is introduced and used to show that a T*(3, 4, v)-code exists for all odd v. Combining this with Levenshtein's result, the existence problem for a T*(3,4, v)-code is solved completely. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 42–53, 2005.
TL;DR: In this paper, all point sets of minimum size in PG(2,q), blocking all external lines to a given irreducible conic, have been determined for every odd q.
Abstract: In a previous paper 1, all point sets of minimum size in PG(2,q), blocking all external lines to a given irreducible conic , have been determined for every odd q. Here we obtain a similar classification for those point sets of minimum size, which meet every external and tangent line to . © 2004 Wiley Periodicals, Inc.
TL;DR: In this paper, the authors describe the generation of all non-orientable triangular embeddings of the complete graphs K12 and K13, and provide an upper bound on the number of simple twofold triple systems of order n.
Abstract: In this paper we describe the generation of all nonorientable triangular embeddings of the complete graphs K12 and K13. (The 59 nonisomorphic orientable triangular embeddings of K12 were found in 1996 by Altshuler, Bokowski and Schuchert, and K13 has no orientable triangular embeddings.) There are 182; 200 nonisomorphic nonorientable triangular embeddings for K12, and 243; 088; 286 for K13. Triangular embeddings of complete graphs are also known as neighborly maps and are a type of twofold triple system. We also use methods of Wilson to provide an upper bound on the number of simple twofold triple systems of order n, and thereby on the number of triangular embeddings of Kn. We discuss applications of our results to exibilit y of embedded graphs.
TL;DR: The necessary conditions for the existence of such a group divisible design are λt(u,t,t;tu) = λ(r(k,r,t), bk(t,k,rtu, k|αtu and α|r), with some definite exceptions.
Abstract: A group divisible design GD(k,λ,t;tu) is α-resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt(u − 1) = r(k − 1), bk = rtu, k|αtu and α|r. It is shown in this paper that these conditions are also sufficient when k = 3, with some definite exceptions. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, it was shown that there is a function g on the natural numbers such that a partial Steiner triple system U on u points can be embedded in a Steiner Triple System V on ν points, in such a way that all automorphisms of U can be extended to V, for every admissible ν satisfying ν>g(u).
Abstract: It is shown that there is a function g on the natural numbers such that a partial Steiner triple system U on u points can be embedded in a Steiner triple system V on ν points, in such a way that all automorphisms of U can be extended to V, for every admissible ν satisfying ν > g(u). We find exponential upper and lower bounds for g. © 2005 Wiley Periodicals, Inc. J Combin Designs.
TL;DR: In this paper, the equivalence between perfect nonlinear functions and appropriate splitting semi-regular relative difference sets was established, and it was shown that there exists a 4-phase perfect non-linear function if and only if the number of input variables is at least twice the output variables.
Abstract: An Erratum has been published for this article in Journal of Combinatorial Designs 14: 82–82, 2006.
We give the equivalence between perfect nonlinear functions and appropriate splitting semi-regular relative difference sets, construct a class of splitting relative difference sets by using Galois rings and bent functions, and prove that there exists a 4-phase perfect nonlinear function if and only if the number of input variables is at least twice the number of output variables. © 2005 Wiley Periodicals, Inc.
TL;DR: In this article, two well-known constructions for Steiner triple systems, recursive and Wilson constructions, were considered and a non-resolvable Wilson construction was given.
Abstract: We consider two well-known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non-resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non-resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005.
TL;DR: This work formulate the construction of cyclic partially balanced incomplete block designs with two associate classes (PBIBD(2)s) as a combinatorial optimization problem and proposes an algorithm based on tabu search to tackle the problem.
Abstract: We formulate the construction of cyclic partially balanced incomplete block designs with two associate classes (PBIBD(2)s) as a combinatorial optimization problem. We propose an algorithm based on tabu search to tackle the problem. Our algorithm constructed 32 new cyclic PBIBD(2)s. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, it was shown that if q is sufficiently large, C is a fixed natural number and |S| = 2q + 2q+1q+C, then roughly 2/3 of the circles of the plane meet S in one point and roughly 1.5 times the number of points in the plane meeting S in four points.
Abstract: Let S be a blocking set in an inversive plane of order q. It was shown by Bruen and Rothschild 1 that |S| ≥ 2q for q ≥ 9. We prove that if q is sufficiently large, C is a fixed natural number and |S = 2q + C, then roughly 2/3 of the circles of the plane meet S in one point and 1/3 of the circles of the plane meet S in four points. The complete classification of minimal blocking sets in inversive planes of order q ≤ 5 and the sizes of some examples of minimal blocking sets in planes of order q ≤ 37 are given. Geometric properties of some of these blocking sets are also studied. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, the existence of (120, 35, 10) difference sets in the 44 nonabelian groups of order 120 has been investigated and it is shown that if a solvable group admits a difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U24
Abstract: Abelian difference sets with parameters (120, 35, 10) were ruled out by Turyn in 1965. Turyn's techniques do not apply to nonabelian groups. We attempt to determine the existence of (120, 35, 10) difference sets in the 44 nonabelian groups of order 120. We prove that if a solvable group admits a (120, 35, 10) difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U24 ≅ 〈x,y : x8 = y3 = 1, xyx−1 = y−1〉. We describe a computer search, which rules out solutions with a ℤ24 quotient. The existence question remains undecided in the three solvable groups admitting a U24 quotient. The question also remains undecided for the three nonsolvable groups of order 120. © 2004 Wiley Periodicals, Inc.
TL;DR: The existence problem of a KHPD( 2u) and a KHCD(2u) is solved with one possible exception of a KhPD(28).
Abstract: A Kirkman holey packing (resp. covering) design, denoted by KHPD(gu) (resp. KHCD(gu)), is a resolvable (gu, 3, 1) packing (resp. covering) design of pairs with u disjoint holes of size g, which has the maximum (resp. minimum) possible number of parallel classes. Each parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3, and 4 when gu ≡ 2, 3, and 4 (mod 3) in turn. In this paper, the existence problem of a KHPD(2u) and a KHCD(2u) is solved with one possible exception of a KHPD(28). © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, the authors present a conjecture that is a common generalization of the Doyen-Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems.
Abstract: In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v ≡ 1,3 (mod 6), u 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v − u). Some results are obtained for v close to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, it was shown that the strategy of constructing a TWH(v) with a prescribed automorphism group cannot succeed with TWh(17), since no 17-player triplewhist tournament has nontrivial automorphisms.
Abstract: The existence of triplewhist tournaments for v players has recently been solved for all values of v except = 17. For v = 12 and v = 13, a complete enumeration has shown the nonexistence of TWh(v), while constructions of TWh(v) have been presented for v > 17. For several values of v, existence has been shown by constructing a TWh(v) with a prescribed, usually cyclic, automorphism group. In this article, it is shown that the strategy of constructing a TWh(v) with a prescribed automorphism group cannot succeed with TWh(17), since no 17-player triplewhist tournament has nontrivial automorphisms. © 2004 Wiley Periodicals, Inc.
TL;DR: The enumeration of regular linear spaces in 1 to at most 19 points is extended and one of the 5 missing cases in the previous list is settled.
Abstract: We extend the enumeration of regular linear spaces in 1 to at most 19 points. In addition, one of the 5 missing cases in the previous list is settled. The number of regular linear spaces of type (15|215,330) is 10,177,328. © 2005 Wiley Periodicals, Inc. J Combin Designs.
TL;DR: This note gives a classification of the self‐dual 𝔽5‐codes of length 48 constructed from the Hadamard matrices of order 24.
Abstract: There are exactly 60 inequivalent Hadamard matrices of order 24 In this note, we give a classification of the self-dual 5-codes of length 48 constructed from the Hadamard matrices of order 24 © 2004 Wiley Periodicals, Inc