Galois ring codes and their images under various bases

Abstract

View the article online for updates and enhancements. Abstract. In this paper we consider linear block codes B of length n over the Galois ring GR(p r , m) and obtain their images with respect to various bases of GR(p r , m) seen as a free module of rank m over the residue class ring Zpr . Interesting new examples of dual, normal and self-dual bases of GR(p r , m) and their relationships are given. The image of B is a linear block code over Zpr of length mn and its generator matrix is formed row-wise by the images of βiG, where {βi} m i=1 is a chosen basis of GR(p r , m) and G is a generator matrix of B. Certain conditions in which the p r -ary image is distance-invariant after a change in basis are investigated. Consequently a new quaternary code endowed with a homogeneous metric that is optimal with respect to certain known bounds is constructed. IntroductionThe motivation for this paper came from finding the p r -ary image of a linear block code B of length n over the Galois ring R = GR(p r , m) with respect to any m-basis of R, and placing a bound on its minimum homogeneous distance in terms of the Hamming distance and other code parameters of B. The p r -ary image is a linear block code of length mn over the residue class ring Z p r , and its distance is with respect to the homogeneous metric defined on GR(p r , m) seen as a finite Frobenius chain ring. The main reference for this result is a previous paper [1] which considers for the most part the so-called polynomial basis of GR(p r , m). It is worthy to investigate the dual and normal bases to see how a change in basis affects the parameters of the p r -ary image. For a thorough discussion of the bases of GR(p r , m) over Z p r the reader is referred to [2]. This paper constructed the dual basis using matrix algebra involving the generalized Frobenius automorphism. It showed that a basis of GR(p r , m) is self-dual if and only its automorphism matrix is orthogonal, and is normal if and only its automorphism matrix is symmetric.The material is organized as follows: Section 2 gives the preliminaries and basic definitions while Section 3 gives the main results. Several new illustrative examples are also provided.