(untitled) (European Journal of Pure and Applied Mathematics, 2013)
We study self-dual codes over the infinite family of rings R_k, which has been recently introduced to the literature. We prove that for each self-dual code over R_k, k ≥ 2,Â there exist a corresponding binary self-dual code, a real unimodular lattice, a complex unimodular lattice, a quaternionic lattice and an infinite family of self-dual codes. We prove the existence of Type II codes of all lengths over R_k, for k≥ 3,Â and we obtain some extremal binary self-dual codes including the extended binary Golay code as the Gray images of self-dual codes over R_k for some suitable k. The binary self-dual codes obtained from R_k all have automorphism groups whose orders are a multiple of 2^k.