Abelian Codes Over Galois Rings Closed Under Certain Permutations

Kiran, T and Rajan, Sundar B (2003) Abelian Codes Over Galois Rings Closed Under Certain Permutations. In: IEEE Transactions on Information Theory, 49 (9). pp. 2242-2253.

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Abstract

We study n-length Abelian codes over Galois rings with characteristic p/sup a/, where n and p are relatively prime, having the additional structure of being closed under the following two permutations: (i) permutation effected by multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and (ii) shifting the coordinates by t positions. A code is t-quasi-cyclic (t-QC) if t is an integer such that cyclic shift of a codeword by t positions gives another codeword. We call the Abelian codes closed under the first permutation as unit-invariant Abelian codes and those closed under the second as quasi-cyclic Abelian (QCA) codes. Using a generalized discrete Fourier transform (GDFT) defined over an appropriate extension of the Galois ring, we show that unit-invariant Abelian and QCA codes can be easily characterized in the transform domain. For t=1, QCA codes coincide with those that are cyclic as well as Abelian. The number of such codes for a specified size and length is obtained and we also show that the dual of an unit-invariant t-QCA code is also an unit-invariant t-QCA code. Unit-invariant Abelian (hence unit-invariant cyclic) and t-QCA codes over Galois field $F_{p^l}$ and over the integer residue rings are obtainable as special cases.

Item Type:	Journal Article
Publication:	IEEE Transactions on Information Theory
Publisher:	IEEE
Additional Information:	©1990 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
Keywords:	Abelian codes;Dual codes;Galois rings;Generalized discrete Fourier transform (GDFT);Mixed-radix number system;Quasi-cyclic codes
Department/Centre:	Division of Electrical Sciences > Electrical Communication Engineering
Date Deposited:	21 Dec 2005
Last Modified:	19 Sep 2010 04:22
URI:	http://eprints.iisc.ac.in/id/eprint/4701

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