Dwijendra K. Ray-Chaudhuri

On resolvable designs

A balanced incomplete block design (BIBD) B[k,@l;@u] is an arrangement of @u elements in blocks of k elements each, such that every pair of elements is contained in exactly @l blocks. A BIBD B[k,1;@u] is called resolvable if the blocks can be petitioned into (@u-1)/(k-1) families each consisting of @u/k mutually disjoint blocks. Ray-Chaudhuri and Wilson [8] proved the existence of resolvable BIBDs B[3,1;@u] for every @u=3 (mod 6). In addition to this result the existence is proved here of resolvable BIBDs B[4,1,@u] for every @u=4 (mod 12).

The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes

One of the most important problems of coding theory is to construct codes with best possible minimum distances. Recently, quasi-cyclic (QC) codes have been proven to contain many such codes. In this paper, we consider quasi-twisted (QT) codes, which are generalizations of QC codes, and their structural properties and obtain new codes which improve minimum distances of best known linear codes over the finite fields GF(3) and GF(5). Moreover, we give a BCH-type bound on minimum distance for QT codes and give a sufficient condition for a QT code to be equivalent to a QC code.

Candelabra systems and designs

Abstract Combinatorial structures called candelabra systems can be used in recursive constructions to build Steiner 3-designs. We introduce a new closure operation on natural numbers involving candelabra systems. This new closure operation makes it possible to generalize various constructions for Steiner 3-designs and to create new infinite families of Steiner 2-designs and 3-designs. We provide an independent proof for Wilsons “product theorem” for Steiner 3-designs. We also construct new group divisible designs of strength 2 and 3.

The Existence of Resolvable Block Designs

Publisher Summary This chapter discusses the existence of resolvable block designs. A block design is a set together with a family of subsets whose members are chosen to satisfy some set of properties that are deemed useful for a particular application. These applications come from many areas, including experimental design, finite geometry, software testing, cryptography, and algebraic geometry. Many variations have been examined, but the most intensely studied are the ones that are balanced incomplete block designs. The balanced incomplete block designs are related to statistical issues in the design of experiments. A block design in which all the blocks have the same size is called uniform.

Constructions of Partial Difference Sets and Relative Difference Sets Using Galois Rings II

In a previous paper, Des., Codes and Cryptogr.8(1996), 215?227]; we used Galois rings to construct partial difference sets, relative difference sets and a difference set. In the present paper, we first generalize and improve the construction of partial difference sets in Des., Codes and Cryptogr.8(1996), 215?227]; also we obtain a family of relative difference sets from these partial difference sets. Second, we construct a class of relative difference sets in (Z4)2m+1?(Z4)r?(Z2?Z2)s,r+s=m, r, s?0 with respect to a subgroup (Z2)2m+1. These constructions make use of character sums from Galois rings, and relate relative difference sets to Hadamard difference sets.

Quasi-cyclic codes over Z/sub 4/ and some new binary codes

Previously, (linear) codes over Z/sub 4/ and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study Z/sub 4/-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to Z/sub 4/ produces a new binary code, a (92, 2/sup 24/, 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials.

A transform approach to permutation groups of cyclic codes over Galois rings

Berger and Charpin (see ibid., vol.42, p.2194-2209, 1996 and Des., Codes Cuyptogr., vol.18, no.1/3, p.29-53, 1999) devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes. We extend this method to cyclic and extended cyclic codes over the Galois ring GR (p/sup a/, m), developing a generalization of the Mattson-Solomon polynomial. In particular, we classify all affine-invariant codes of length 2/sup m/ over Z/sub 4/, thus generalizing the corresponding result of Kasami, Lin, and Peterson (1967) and giving an alternative proof to Abdukhalikov. We give a large class of codes over Z/sub 4/ with large permutation groups, which include generalizations of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Muller (RM) codes.

Determination of all possible orders of weight 16 circulant weighing matrices

We show that a circulant weighing matrix of order n and weight 16 exists if and only if n>=21 and n is a multiple of 14,21 or 31.

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z4 × Z4 × Z6.

On existence and number of orthogonal arrays

Abstract It is shown that a signed orthogonal array SAt(v, k, λ) exists for all (v, k, λ, t), k ⩾ t and an orthogonal array At(v, k, λ) exists for all (v, k, λ, t) provided λ is sufficiently large. A reciprocity relation for the number of distinct orthogonal arrays is derived as in the case of designs in a recent paper of Singhi and Shrikhande.