Matrix-Product Codes over ? q (original) (raw)

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|2_u+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.

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Authors and Affiliations

  1. Infineon Technologies, Stoke Gifford, BS34 8HP, UK (e-mail: tim.blackmore@infineon.com), , , , , , GB
    Tim Blackmore
  2. Department of Mathematics, University of Queensland, Brisbane 4072, Australia (e-mail: ghn@maths.ug.edu.au), , , , , , AU
    Graham H. Norton

Authors

  1. Tim Blackmore
  2. Graham H. Norton

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Received: July 20, 1999; revised version: August 27, 2001

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Blackmore, T., Norton, G. Matrix-Product Codes over ? q .AAECC 12, 477–500 (2001). https://doi.org/10.1007/PL00004226

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