Matrix-Product Codes over ? q

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.