Old and new designs via difference multisets and strong difference families

Let X =((x_1,1,x_1,2,…,x_1,k),(x_2,1,x_2,2,…,x_2,k),…,(x_t,1,x_t,2,…,x_t,k)) be a family of t multisets of size k defined on an additive group G. We say that X is a t-(G,k,μ) strong difference family (SDF) if the list of differences (x_h,i-x_h,j∣h=1,…,t;i≠ j) covers all of G exactly μ times. If a SDF consists of a single multiset X, we simply say that X is a (G,k,μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1-rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1-rotational (in many cases resovable) BIBD's.     

Existence of (q,k, 1) difference families with q a prime power and k = 4, 5

The existence of a (q, k, 1) difference family in GF(q) has been completely solved for k = 3. For k = 4, 5 partial results have been given by Bose, Wilson, and Buratti. In this article, we continue the investigation and show that the necessary condition for the existence of a (q, k, 1) difference family in GF(q), i.e., q ≡ 1 (mod k(k − 1)) is also sufficient for k = 4, 5. For general k, Wilson's bound shows that a (q, k, 1) difference family in GF(q) exists whenever q ≡ 1 (mod k(k − 1)) and q > [k(k − 1)/2]^k(k−1). An improved bound on q is also presented. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 21–30, 1999     

Recursive constructions for difference matrices and relative difference families

We present a new recursive construction for difference matrices whose application allows us to improve some results by D. Jungnickel. For instance, we prove that for any Abelian p-group G of type (n₁, n₂, …, n_t) there exists a (G, p^e, 1) difference matrix with e = Also, we prove that for any group G there exists a (G, p, 1) difference matrix where p is the smallest prime dividing |G|. Difference matrices are then used for constructing, recursively, relative difference families. We revisit some constructions by M. J. Colbourn, C. J. Colbourn, D. Jungnickel, K. T. Phelps, and R. M. Wilson. Combining them we get, in particular, the existence of a multiplier (G, k, λ)-DF for any Abelian group G of nonsquare-free order, whenever there exists a (p, k, λ)-DF for each prime p dividing |G|. Then we focus our attention on a recent construction by M. Jimbo. We improve this construction and prove, as a corollary, the existence of a (G, k, λ)-DF for any group G under the same conditions as above. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 165–182, 1998     

Existence of (q, 7, 1) difference families with q a prime power

The existence of a (q,k, 1) difference family in GF(q) has been completely solved for k = 3,4,5,6. For k = 7 only partial results have been given. In this article, we continue the investigation and use Weil's theorem on character sums to show that the necessary condition for the existence of a (q,7,1) difference family in GF(q), i.e. q ≡ 1; (mod 42) is also sufficient except for q = 43 and possibly except for q = 127, q = 211, q = 31⁶ and primes q∈ [261239791, 1.236597 × 10¹³] such that in GF(q). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 126–138, 2002; DOI 10.1002/jcd.998     

A New Class of Optimal 3-splitting Authentication Codes

The necessary condition for the existence of a (ν, 3× 3,1)-splitting BIBD is ν ≡ 1 (mod 54). In this paper, we show that the necessary condition is also sufficient with one possible exception of ν = 55. As its application, we obtain a new infinite class of optimal 3-splitting authentication codes. AMS Classification: 05B05, 94A62 An erratum to this article is available at .     

Constructing difference families through an optimization approach: Six new BIBDs

In this paper we formulate the construction of difference families as a combinatorial optimization problem. A tabu search algorithm is used to find an optimal solution to the optimization problem for various instances of difference families. In particular, we construct six new difference families which lead to an equal number of new balanced incomplete block designs with parameters: (49, 98, 18, 9, 3), (61, 122, 20, 10, 3), (46, 92, 20, 10, 4), (45, 90, 22, 11, 5), (85, 255, 24, 8, 2) and (34, 85, 30, 12, 10). © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 261–273, 2000     

Realizations for direct constructions of resolvable Steiner 2‐designs with block size 5

We consider direct constructions due to R. J. R. Abel and M. Greig, and to M. Buratti, for ({ν},5,1) balanced incomplete block designs. These designs are defined using the prime fields F_p for certain primes p, are 1‐rotational over G ⊕ F_p where G is a group of order 4, and are also resolvable under certain conditions. We introduce specifications to the constructions and, by means of character sum arguments, show that the constructions yield resolvable designs whenever p is sufficiently large. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:207–217, 2000     

Existence of (q,6,1) Difference Families withq a Prime Power

The existence of a (q,k,1) difference family in GF(q) has been completely solved for k=3, 4, 5. For k=6 fundamental results have been given by Wilson. In this article, we continue the investigation and show that the necessary condition for the existence of a(q,6,1) difference family in GF(q), i.e. q 1 (mod 30) is also sufficient with one exception of q=61. The method of this paper is to lower Wilson's bound by using Weil's theorem on character sums to exploit Wilson's sufficient conditions for the existence of (q,6,1) difference families. The remaining gap is closed by computer searches.     

A construction of balanced arrays of strengtht and some related incomplete block designs

Summary Saha [6] has shown the equivalence between a ‘tactical system’ (or at-design) and a 2-symbol balanced array (BA) of strengtht. The implicit method of construction of BA in that paper has been generalized herein to that of ans-symbol BA of strengtht. Some BIB and PBIB designs are also constructed from these arrays. Majindar [2], Vanstone [8] and Saha [6] have all shown that the existence of a symmetrical BIBD forv treatments implies the existence of six more BIBD's forv treatments in (v/2) blocks. An analogue of this result has been obtained for a large class of PBIB designs in this paper.     

A complete classification of (12,4,3)‐RBIBDs

Using a parallel class by parallel class backtracking algorithm we classify (12,4,3)‐RBIBDs. The size of the search space is reduced by combinatorial arguments. Five nonisomorphic RBIBDs are found and listed. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 385–400, 2001     

(C_k \oplus G,k,\lambda ) Difference Families

Let G be an additive group and C _k be the additive group of the ring Z _k of residues modulo k. If there exist a (G, k, ) difference family and a (G, k, ) perfect Mendelsohn difference family, then there also exists a difference family. If the (G, k, ) difference family and the (G, k, ) perfect Mendelsohn difference family are further compatible, then the resultant difference family is elementary resolvable. By first constructing several series of perfect Mendelsohn difference families, many difference families and elementary resolvable difference families are thus obtained.     

There exists no (15,5,4) RBIBD

An (n, M, d)_q code is a q‐ary code of length n, cardinality M, and minimum distance d. We show that there exists no (15,5,4) resolvable balanced incomplete block design (RBIBD) by showing that there exists no (equidistant) (14,15,10)₃ code. This is accomplished by an exhaustive computer search using an orderly algorithm combined with a maximum clique algorithm. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 227–232, 2001     

There exists no (15,5,4) RBIBD†*

An (n, M, d)_q code is a q‐ary code of length n, cardinality M, and minimum distance d. We show that there exists no (15,5,4) resolvable balanced incomplete block design (RBIBD) by showing that there exists no (equidistant) (14,15,10)₃ code. This is accomplished by an exhaustive computer search using an orderly algorithm combined with a maximum clique algorithm. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 357–362, 2001     

The spectrum of (<Emphasis Type="Italic">gv</Emphasis>, <Emphasis Type="Italic">g</Emphasis>, 3, λ)-DF in <Emphasis Type="Italic">Z</Emphasis><Subscript><Emphasis Type="Italic">gv</Emphasis></Subscript>

In this paper, the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)-difference family in Z _gv are established. As a consequence, the existence spectrum of a cyclic (3, λ)-GDD of type g ^v without short orbits is determined. This work was supported by National Natural Science Foundation of China (Grants Nos. 10771013, 10831002)     

A Construction for Resolvable Designs and Its Generalizations

 In [14], D.K. Ray-Chaudhuri and R.M. Wilson developed a construction for resolvable designs, making use of free difference families in finite fields, to prove the asymptotic existence of resolvable designs with index unity. In this paper, generalizations of this construction are discussed. First, these generalizations, some of which require free difference families over rings in which there are some units such that their differences are still units, are used to construct frames, resolvable designs and resolvable (modified) group divisible designs with index not less than one. Secondly, this construction method is applied to resolvable perfect Mendelsohn designs. Thirdly, cardinalities of such sets of units are investigated. Finally, composition theorems for free difference families via difference matrices are described. They can be utilized to produce some new examples of resolvable designs.     

Constructions for optimal (u×v,{3,4},1,Q)-OOSPCs

Multiple-weight optical orthogonal signature pattern codes (OOSPCs) were introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirement. In this paper, an upper bound on the maximum code size of a $(u\times v,W,\lambda ,Q)$-OOSPC is obtained. A link between optimal $(u\times v,W,\lambda ,Q)$-OOSPCs and block designs is developed. Several infinite families of optimal $(u\times v,\{3,4\},1,Q)$-OOSPCs are presented by means of semi-cyclic group divisible designs ($(W,Q)$-SCGDDs) and perfect relative difference families.     

相对差集和组合集的构造

利用Galois环、Bent函数、Gaolis环上的部分指数和等技巧,构造了指数不超过4的有限交换群上的分裂型相对差集和一类非分裂型组合集.     

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ‐BIBDs are (mod ) and . In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ‐RBIBD, namely (mod ) and , are sufficient. A few new pairwise balanced designs are also given.     

Uniform supersaturated design and its construction

Supersaturated designs are factorial designs in which the number of main effects is greater than the number of experimental runs. In this paper, a discrete discrepancy is proposed as a measure of uniformity for supersaturated designs, and a lower bound of this discrepancy is obtained as a benchmark of design uniformity. A construction method for uniform supersaturated designs via resolvable balanced incomplete block designs is also presented along with the investigation of properties of the resulting designs. The construction method shows a strong link between these two different kinds of designs     

区传递的$2-(v, k, 1)$ 设计与单群$E_6(q)$

This article is a contribution to the study of block-transitive automorphism groups of 2-（v,k,1） block designs. Let D be a 2-（v,k,1） design admitting a block-transitive, pointprimitive but not flag-transitive automorphism group G. Let kr = （k,v-1） and q = pf for prime p. In this paper we prove that if G and D are as above and q （3（krk-kr ＋ 1）f）1/3, then G does not admit a simple group E6（q） as its socle.