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     

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.     

New constructions for perfect hash families and related structures using combinatorial designs and codes

In this paper, we consider explicit constructions of perfect hash families using combinatorial methods. We provide several direct constructions from combinatorial structures related to orthogonal arrays. We also simplify and generalize a recursive construction due to Atici, Magliversas, Stinson and Wei [3]. Using similar methods, we also obtain efficient constructions for separating hash families which result in improved existence results for structures such as separating systems, key distribution patterns, group testing algorithms, cover‐free families and secure frameproof codes. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:189–200, 2000     

Strong difference families,difference covers,and their applications for relative difference families

In (M. Buratti, J Combin Des 7:406–425, 1999), Buratti pointed out the lack of systematic treatments of constructions for relative difference families. The concept of strong difference families was introduced to cover such a problem. However, unfortunately, only a few papers consciously using the useful concept have appeared in the literature in the past 10 years. In this paper, strong difference families, difference covers and their connections with relative difference families and optical orthogonal codes are discussed.      

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.     

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     

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.     

A Padé family of iterations for the matrix sign function and related problems

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p‐sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p‐sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1?z)^?σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log‐concavity property. Copyright © 2011 John Wiley & Sons, Ltd.