Kevin T. Phelps

On binary 1-perfect additive codes: some structural properties

The rank and kernel of 1-perfect additive codes is determined. Additive codes could be seen as translation-invariant propelinear codes and, in this correspondence, a characterization of propelinear codes as codes having a regular subgroup of the full group of isometrics of the code is established. A characterization of the automorphism group of a 1-perfect additive code is given and also the cardinality of this group is computed. Finally, an efficiently computable characterization of the Steiner triple systems associated with a 1-perfect binary additive code is also established.

Kernels of nonlinear Hamming codes

Given a binary code,C, of lengthn, thekernel of the code is defined to be the set of all vectors which leave the code invariant under translation. Throughout this paper, various properties of kernels will be considered. However, the main idea of this paper is to show the necessary and sufficient conditions for the existence of kernels of all possible sizes for nonlinear perfect binary codes.

On the additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes: rank and kernel

All the possible nonisomorphic additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one.

Constant Weight Codes and Group Divisible Designs

The study of a class of optimal constant weight codes over arbitrary alphabets was initiated by Etzion, who showed that such codes are equivalent to special GDDs known as generalized Steiner systems GS(t,k,n,g) Etzion. This paper presents new constructions for these systems in the case t=2, k=3. In particular, these constructions imply that the obvious necessary conditions on the length n of the code for the existence of an optimal weight 3, distance 3 code over an alphabet of arbitrary size are asymptotically sufficient.

Quaternary Reed-Muller codes

The class of quaternary Reed-Muller codes is introduced as a generalization of Reed-Muller codes to /spl Zopf//sub 4/-linear codes. Properties of this class of codes are established including the rank and kernel of the Gray map image of codes in this class. The class includes all /spl Zopf//sub 4/-linear Kerdock-like and Preparata-like codes.

On Perfect Codes: Rank and Kernel

We establish upper and lower bounds on the rank and the dimension of the kernel of perfect binary codes. We also establish some results on the structure of perfect codes.

The rank and kernel of extended 1-perfect Z 4 -linear and additive non-Z 4 -linear codes

A binary extended 1-perfect code of length n + 1 = 2/sup t/ is additive if it is a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//. The punctured code by deleting a /spl Zopf//sub 2/ coordinate (if there is one) gives a perfect additive code. 1-perfect additive codes were completely characterized and by using that characterization we compute the possible parameters /spl alpha/, /spl beta/, rank, and dimension of the kernel for extended 1-perfect additive codes. A very special case is that of extended 1-perfect /spl Zopf//sub 4/-linear codes.

Constructions of perfect Mendelsohn designs

Abstract Let n and k be positive integers. An (n, k, 1)-Mendelsohn design is an ordered pair (V, C ) where V is the vertex set of Dn, the complete directed graph on n vertices, and C is a set of directed cycles (called blocks) of length k which form an arc-disjoint decomposition of Dn. An (n, k, 1)-Mendelsohn design is called a perfect design and denoted briefly by (n, k, 1)-PMD if for any r, 1⩽r⩽k-1, and for each (x, y) ϵ V×V there is exactly one cycle c∈ C in which the (directed) distance along c from x to y is r. A necessary condition for the existence of an (n, k, 1)-PMD is n(n-1)≡0(modk). In this paper we shall describe some new techniques used in the construction of PMDs, including constructions of the product type. As an application, we show that the necessary condition for the existence of an (n, 5, 1)-PMD is also sufficient, except for n=6 and with at most 21 possible exceptions of n of which 286 is the largest.

On /spl Zopf//sub 4/-linear Preparata-like and Kerdock-like codes

We say that a binary code of length n is additive if it is isomorphic to a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /spl alpha/ + 2/spl beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /spl alpha/ = 0, i.e., it is always a /spl Zopf//sub 4/-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the /spl Zopf//sub 4/-dual of these codes, i.e., the /spl Zopf//sub 4/-linear Kerdock-like codes.

The spectrum for 2-perfect 6-cycle systems

Abstract Recently, the spectrum problem for 2-perfect m -cycle systems has been studied by several authors. In this paper we find the spectrum for 2-perfect 6-cycle systems with two possible exceptions. The connection between these systems and quasigroups satisfying some 2 variable identities is discussed.