Optimum commutative group codes (Q2256103)

A code \(C(M,n,d)\) with a given order \(M\), dimension \(n\) and minimum distance \(d\) is said to be optimum, if \(d\) is the largest minimum distance for a fixed \(M\) and \(n\). After a preliminary presentation of commutative group codes and some of their properties the authors discuss the initial vector problem for those codes and they characterize it as a linear programming problem. Then the main results of this paper are presented: the authors prove an extended Hermite normal form, which allows the characterization of isometric lattices by coordinate permutation. Also two theorems are proved, which provide a significant reduction in the number of codes to be checked when searching an optimum one. In the end of this paper the authors present their method as a pseudo-code and they give also many examples of optimum commutative group codes.