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^m, 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 Z₄-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 Z₂^k1⊕Z₄^k2⊕Q ₈(k3) where Q₈ 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 Z₂^k1⊕Z₄^k2⊕Q ₈^k3 with k₁+2k₂+4k₃ =n. For k₂=k₃=0 we obtain linear binary codes and for k₁=k₃=0 we obtain Z₄-linear codes. The class of additive propelinear codes-the Abelian 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