Maximum Distance Separable (MDS) Matrix of size m x m over Zq

Abstract

The Maximum Distance Separable (MDS) code is one of the codes that known as error-correcting code where the generator matrix [I|A] is arranged by the identity matrix and the MDS matrix. In coding, MDS matrix can detect and correct errors optimally. A matrix over the Zq is called an MDS matrix if and only if all the determinants of its square submatrix are non-zero. A matrix over the Zq is called an MDS matrix if and only if all the determinants of its square submatrix are non-zero. In m x m matrix over Zq, the analyzed of possible entries and determinants of submatrix can be declare the existence of an MDS matrix of size m x m over Zq. The result is there will be no MDS matrix of size m x m where m greater than or equal to [(q-1)^2 + 1] - [q-2] for Zq with any of q. For Zq  with q prime, there will be no MDS matrix of size m x m where m greater than or equal to [(q-1)^2 + 1] - [q-2] - [1/2 x (q-1)].