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

Katz, Daniel J. (2006) p-Adic estimates of Hamming weights in Abelian codes over Galois rings. IEEE Transactions on Information Theory, 52 (3). pp. 964-985. ISSN 0018-9448. https://resolver.caltech.edu/CaltechAUTHORS:KATieeetit06

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Abstract

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/TIT.2005.864428DOIUNSPECIFIED
Additional Information:© Copyright 2006 IEEE. Reprinted with permission. Manuscript received September 8, 2004; revised September 21, 2005. [Posted online: 2006-03-06] This work was supported in part by the Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics at Caltech, given by Steve and Rosemary Johnson. Communicated by A. E. Ashikhmin, Associate Editor for Coding Theory. The author wishes to thank R. M. Wilson and R. J. McEliece for their interest and support. He also thanks the Associate Editor and anonymous referees for helpful suggestions which improved this paper.
Subject Keywords:Abelian codes, codes over Galois rings, counting polynomials, McEliece’s theorem.
Issue or Number:3
Record Number:CaltechAUTHORS:KATieeetit06
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:KATieeetit06
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3411
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:06 Jun 2006
Last Modified:02 Oct 2019 23:03

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