Self-Dual Codes over R_k and Binary Self-Dual Codes

Steven T. Dougherty, Bahattin Yıldız, Suat Karadeniz

Abstract

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 \geq 2,  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\geq 3,  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.

Keywords

self-dual codes; codes over R_k; extremal codes; local rings

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