Double circulant self-dual codes over ℤ _2k

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 ℤ ₆, ℤ ₈, ℤ ₁₀, and ℤ ₁₂.