Algebraic characterization of MDS group codes over cyclic groups

An (n,k) group code over a group G is a subset of Gⁿ which forms a group under componentwise group operation and can be defined in terms of n-k homomorphisms from G^k to G. The set of homomorphisms which define maximum distance separable (MDS) group codes defined over cyclic groups are characterized. Each defining homomorphism can be specified by a set of k endomorphisms of G. A matrix is associated with the k(n-k) defining endomorphisms of the code and necessary and sufficient conditions for this matrix to define an MDS code over cyclic groups is proved. Using this matrix characterization it is proved that over a cyclic group with M elements, where M=p₁^d(1)p₂^d(2) ···p_m^d(m),(k+s,k) MDS group codes, for all s,k⩾2, do not exist if max {s,k}⩾min {p₁,p₂,··,p_m}. Finally, it is shown that the dual code of an MDS group code over C_M, a cyclic group with M elements, is also an MDS group code