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Codes Closed under Arbitrary Abelian Group of Permutations

Dey, Bikash Kumar and Rajan, Sundar B (2004) Codes Closed under Arbitrary Abelian Group of Permutations. In: SIAM Journal on Discrete Mathematics, 18 (1). pp. 1-18.

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Abstract

Algebraic structure of codes over $F_{q}$, closed under arbitrary abelian group G of permutations with exponent relatively prime to q, called G-invariant codes, is investigated using a transform domain approach. In particular, this general approach unveils algebraic structure of quasicyclic codes, abelian codes, cyclic codes, and quasi-abelian codes with restriction on G to appropriate special cases. Dual codes of G-invariant codes and self-dual G-invariant codes are characterized. The number of G-invariant self-dual codes for any abelian group G is found. In particular, this gives the number of self-dual l-quasi-cyclic codes of length ml over $ F_{q}$ when (m, q) = 1. We extend Tanner’s approach for getting a bound on the minimum distance from a set of parity check equations over an extension field and outline how it can be used to get a minimum distance bound for a G-invariant code. Karlin’s decoding algorithm for a systematic quasi-cyclic code with a single row of circulants in the generator matrix is extended to the case of systematic quasi-abelian codes. In particular, this can be used to decode systematic quasi-cyclic codes with columns of parity circulants in the generator matrix.

Item Type: Journal Article
Additional Information: The copyright of this article belongs to Society for Industrial and Applied Mathematics (SIAM).
Keywords: quasi-cyclic codes;permutation group of codes;discrete Fourier transform;self-dual codes
Department/Centre: Division of Electrical Sciences > Electrical Communication Engineering
Date Deposited: 20 Nov 2007
Last Modified: 19 Sep 2010 04:17
URI: http://eprints.iisc.ernet.in/id/eprint/2310

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