Double circulant self-dual codes over Z2k

Recently there has been tremendous interest in self-dual codes over finite rings, specifically the rings Z_2k. In this paper, we investigate double circulant self-dual codes over Z_2k for small k and short lengths. In particular, we give a classification of the extremal double circulant codes. Using invariant theory, a basis for the space of invariants to which the Hamming weight enumerators belong is given for self-dual codes over Z₆, Z₈, Z₁₀ , and Z₁₂