Translation-invariant propelinear codes

A class of binary group codes is investigated. These codes are the propelinear codes, defined over the Hamming metric space Fn, 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 Z2k1 ⊕ Z4k2 ⊕ Q8k3 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 Z2k1 ⊖ Z4k2 ⊕ Q8k3 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. © 1997 IEEE.