Matrix-Product Codes over ?
q |
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Authors: | Tim Blackmore Graham H Norton |
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Affiliation: | (1) Infineon Technologies, Stoke Gifford, BS34 8HP, UK (e-mail: tim.blackmore@infineon.com), GB;(2) Department of Mathematics, University of Queensland, Brisbane 4072, Australia (e-mail: ghn@maths.ug.edu.au), AU |
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Abstract: | Codes C
1
,…,C
M
of length n over ?
q
and an M × N matrix A over ?
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
⊥. 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
⊥ can be described using C
1
⊥, …,C
M
⊥ and a transformation of A. This yields d(C
⊥). Finally we show that an NSC matrix-product code is a generalized concatenated code.
Received: July 20, 1999; revised version: August 27, 2001 |
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Keywords: | Binary (u|u + v)-construction Ternary (u + v + w|2u + v|u)-construction Generalized Reed-Muller Codes Generalized concatenated codes |
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