On the algebraic structure of quasi-cyclic codes. II: Chain rings
type: Item
instance of: scholarly article
title: On the algebraic structure of quasi-cyclic codes. II: Chain rings
zbMATH Open document ID: 1032.94011
author: San Ling
author: Patrick Solé
published in: Designs, Codes and Cryptography
publication date: 2003-09-07T00:00:00Z
review text: This paper is Part II of an investigation of the algebraic structure of quasi-cyclic (QC) codes. QC codes over a finite commutative ring \(A\) are viewed as codes over the polynomial ring \[ R(A,m) := A[Y]/(Y^m-1). \] This latter ring can be decomposed into a direct sum of local rings. This can be achieved either by the Chinese Remainder Theorem or the Discrete Fourier Transform. The ring decomposition technique of Part I [IEEE Trans. Inf. Theory 47, 2751-2760 (2001; Zbl 1023.94015)] is extended to the case when the factors in the direct product decomposition are no longer fields but arbitrary chain rings. This includes not only the case of quasi-cyclic codes over rings but also the case of quasi-cyclic codes over fields whose co-index is no longer prime to the characteristic field. This approach allows for a systematic investigation of self-dual QC codes. In addition many well-known code constructions are obtained, as well as new ones (quinting and septing). Several examples are given, including the binary and ternary extended Golay codes. A new quaternary construction of the Leech lattice is derived.
reviewed by: T. Aaron Gulliver
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94B15
zbMATH DE Number: 1975931
zbMATH Keywords: code construction
zbMATH Keywords: quasi-cyclic codes
zbMATH Keywords: DFT
zbMATH Keywords: CRT
zbMATH Keywords: chain rings
zbMATH Keywords: self-dual codes
zbMATH Keywords: Leech lattice
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1975931