Note on MDS codes over the integers modulo pm

where C is a code of length n over A , k=\log_{|A|}|C| and d(C) is the minimum distance of C and proved several nonexistence results for MDS group codes over finite groups with respect to the above bound, that is, the group codes with d(C)=n-k+1 . Zain and Rajan [9] also proved that for a group code C over a cyclic group of m elements with generator matrix of the form (I_{k}|M) , where M is a k\cross(n-k) matrix over \mathbb{Z}_{m} , C is MDS iff the determinant of every h\cross h submatrix, h=1,2 , \ldots , \min\{n-k, k\} , of M is a unit in \mathbb{Z}_{m} . Moreover, Dong, Soh and Gunawan [3] proved a similar matrix characterization of MDS (free) codes with parity check matrices of the form (-M|I_{n-k}) over modules. Recently, Shiromoto and Yoshida [8] introduced a Singleton bound for linear codes over \mathbb{Z}_{k} as follows: