On the construction of group block codes

We consider the construction of group block codes, i.e., subgroups of Gⁿ, the n-fold direct product of a group G. Two concepts are introduced that make this construction similar to that of codes over gf(2). The first concept is that of an indecomposable code. The second is that of a parity-check matrix. As a result, group block codes over a decomposable Abelian group of exponent d_m can be seen as block codes over the ring of residues modulo d_m, and their minimum Hamming distance can be easily determined. We also prove that, under certain technical conditions, (n, k) systematic group block codes over non-Abelian groups are asymptotically bad, in the sense that their minimum Hamming distance cannot exceed n/k].