On the Hamming distance of linear codes over a finite chain ring

Let R be a finite chain ring (e.g. a Galois ring), K its residue field and C a linear code over R. We prove that d(C), the Hamming distance of C, is d((C : α)), where (C : α) is a submodule quotient, α is a certain element of R and — denotes the canonical projection to K. These two codes also have the same set of minimal codeword supports. We explicitly construct a generator matrix/polynomial of (C : α) from the generator matrix/polynomials of C. We show that in general d(C) ≤ d(C) with equality for free codes (i.e. for free R- submodules of Rn) and in particular for Hensel lifts of cyclic codes over K. Most of the codes over rings described in the literature fall into this class. We characterise MDS codes over R and prove several analogues of properties of MDS codes over finite fields. We compute the Hamming weight enumerator of a free MDS code over R

A Class of Three-Weight Cyclic Codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of three-weight cyclic codes over \gf(p) whose duals have two zeros is presented, where $p$ is an odd prime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page

p-Adic estimates of Hamming weights in Abelian codes over Galois rings

A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more