Matrix-product codes over Fq

Blackmore, Tim and Norton, Graham H. (2001) Matrix-product codes over Fq. Applicable Algebra In Engineering Communication And Computing, 12 6: 477-500. doi:10.1007/PL00004226

Author Blackmore, Tim
Norton, Graham H.
Title Matrix-product codes over Fq
Journal name Applicable Algebra In Engineering Communication And Computing   Check publisher's open access policy
ISSN 0938-1279
Publication date 2001-12
Sub-type Article (original research)
DOI 10.1007/PL00004226
Volume 12
Issue 6
Start page 477
End page 500
Total pages 24
Editor J. Calmet
Place of publication Heidelberg, Germany
Publisher Springer-Verlag
Collection year 2001
Language eng
Subject C1
230103 Rings And Algebras
780101 Mathematical sciences
Abstract Codes C-1,...,C-M of length it over F-q and an M x N matrix A over F-q define a matrix-product code C = [C-1 (...) C-M] (.) A consisting of all matrix products [c(1) (...) c(M)] (.) A. This generalizes the (u/u + v)-, (u + v + w/2u + v/u)-, (a + x/b + x/a + b + x)-, (u + v/u - v)- etc. constructions. We study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of /C/, d(C), the minimum Hamming distance of C, and C-perpendicular to. It also reveals an interesting connection with MDS codes. We determine /C/ when A is non-singular. To underbound d(C), we need A to be 'non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary 'Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C-perpendicular to can be described using C-1(perpendicular to),...,C-M(perpendicular to) and a transformation of A. This yields d(C-perpendicular to). Finally we show that an NSC matrix-product code is a generalized concatenated code.
Keyword Computer Science, Interdisciplinary Applications
Computer Science, Theory & Methods
Mathematics, Applied
Binary (u/u Plus V)-construction
Ternary (u Plus V + W Vertical Bar 2u+v Vertical Bar U)-construction
Generalized Reed-muller Codes
Generalized Concatenated Codes
Q-Index Code C1

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
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Created: Tue, 14 Aug 2007, 15:03:03 EST