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

Affine invariant extended cyclic codes over galois rings

Recently, Blackford and Ray-Chaudhuri used transform domain techniques to permutation groups of cyclic codes over Galois rings. They used the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length 2m over any subring of GR(4,m) to be affine invariant. Here, we use the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length pm over any subring of GR(pe,m) to be affine invariant, for e=2 with arbitrary p and for p=2 with arbitrary e. These are used to find two new classes of affine invariant Bose-Chaudhuri-Hocquenghem (BCH) and generalized Reed-Muller (GRM) codes over Z2e for arbitrary e and a class of affine invariant BCH codes over Zp2 for arbitrary prime p.© IEE

Affine Invariant Extended Cyclic Codes over Galois Rings

Afflne invariant extended cyclic codes of length $p^m$ over any subring of $GR(p^e,m)$ for e = 2 with arbitrary p and for p = 2 with arbitrary e are characterized using transform technique. New classes of afflne invariant BCH and GRM codes over such rings are found