Matrix characterization of MDS linear codes over modules Original Research Article

Let R be a commutative ring with identity, N be an R-module, and M = (aij)r×k be a matrix over R. A linear code C of length n over N is defined to be a submodule of Nn. It is shown that a linear code C(k, r) with parity check matrix (−M¦Ir) is maximum distance separable (MDS) iff the determinant of every h × h submatrix, h = 1, 2,…, min{k, r}, of M is not an annihilator of any nonzero element of N. This characterization is used to derive some results for group codes over abelian groups.