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p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/)

Katz, Daniel J. (2005) p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/). IEEE Transactions on Information Theory, 51 (1). pp. 281-305. ISSN 0018-9448.

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Counting polynomial techniques introduced by Wilson are used to provide analogs of a theorem of McEliece. McEliece's original theorem relates the greatest power of p dividing the Hamming weights of words in cyclic codes over GF (p) to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over /spl Zopf/(2/sup d/) using various weight functions (Hamming, Lee, and Euclidean weight as well as count of occurrences of a particular symbol). Some of these results were strengthened by Wilson, who also considered the alphabet /spl Zopf/(p/sup d/) for p an arbitrary prime. These previous results, new strengthened versions, and generalizations are proved here in a unified and comprehensive fashion for the larger class of Abelian codes over /spl Zopf/(p/sup d/) with p any prime. For Abelian codes over /spl Zopf//sub 4/, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McEliece's theorem obtained by Wilson (for Hamming weight, Lee weight, and symbol counts) and the analog obtained here for Euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose Fourier transforms have a specified support.

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Additional Information:© Copyright 2006 IEEE. Reprinted with permission. Manuscript received April 27, 2004; revised September 24, 2004. [ p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/) Katz, D.J. Dept. of Math., California Inst. of Technol., Pasadena, CA, USA This paper appears in: Information Theory, IEEE Transactions on Publication Date: Jan. 2005 Volume: 51 , Issue: 1 On page(s): 281 - 305 ISSN: 0018-9448 INSPEC Accession Number:8267907 Digital Object Identifier: 10.1109/TIT.2004.839495 Posted online: 2005-01-10] 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 R. J. McEliece, Associate Editor for Coding Theory. The author would like to thank R. M. Wilson for introducing him to this area of research and for much advice and support. The author also wishes to thank R. J. McEliece for his interest, support, and for drawing his attention to [15], which greatly informed and influenced these researches. Finally, the author thanks an anonymous referee for a careful reading of the manuscript and for the useful suggestion that guiding examples on counting polynomials and the Main Theorem be included.
Subject Keywords:Abelian codes, codes over rings, counting polynomials, McEliece’s theorem
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Record Number:CaltechAUTHORS:KATieeetit05
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ID Code:3410
Deposited By: Archive Administrator
Deposited On:06 Jun 2006
Last Modified:02 Oct 2019 23:02

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