MATRIX CHARACTERIZATION OF MDS LINEAR CODES OVER MODULES

Let R be a commutative ring with identity, N be an R-module, and M = ( a(ij))(rxk) be a matrix over R, A linear code C of length n over N is defined to be a submodule of N-n. It is shown that a linear code C(k, r) with parity check matrix (-M/I-r,) is maximum distance separable (M DS) iff the determinant of every hxh submatrix, h = 1,2,...:min(k, r), of M is not an annihilator of any nonzero element of N. This characte rization is used to derive some results for group I.odes over abelian groups. (C) 1998 Elsevier Science Inc. All rights reserved.