Translation-invariant propelinear codes

A class of binary group codes is investigated. These codes are the propelinear codes, defined over the Hamming metric space [F.sup.n], F = {0, 1}, with a group structure. Generally, they are neither Abelian nor translation-invariant codes but they have good algebraic and combinatorial properties. Linear codes and [Z.sub.4]-linear codes can be seen as a subclass of propelinear codes. It is shown here that the subclass of translation-invariant propelinear codes is of type [Mathematical Expression Omitted] where [Q.sub.8] is the non-Abelian quaternion group of eight elements. Exactly, every translation-invariant propelinear code of length n can be seen as a subgroup of [Mathematical Expression Omitted] with [k.sub.1] + 2[k.sub.2] + 4[k.sub.3] = n. For [k.sub.2] = [k.sub.3] = 0 we obtain linear binary codes and for [k.sub.1] = [k.sub.3] = 0 we obtain [Z.sub.4]-linear codes. The class of additive propelinear codes - the Abellan subclass of the translation-invariant propelinear codes - is studied and a family of nonlinear binary perfect codes with a very simply construction and a very simply decoding algorithm is presented. Index Terms - Propelinear codes, translation-invariant propelinear codes, additive codes, perfect codes, 24-linear codes, [Q.sub.8]codes.