Abstract
Using ideas from the cohomology of finite groups, an isomorphism is established between a group ring and the direct sum of twisted group rings. This gives a decomposition of a group ring code into twisted group ring codes. In the abelian case the twisted group ring codes are (multi-dimensional) constacyclic codes. We use the decomposition to prove that, with respect to the Euclidean inner product, there are no self-dual group ring codes when the group is the direct product of a 2-group and a group of odd order, and the ring is a field of odd characteristic or a certain modular ring. In particular, there are no self-dual abelian codes over the rings indicated. Extensions of these results to non-Euclidean inner products are briefly discussed.
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Hughes, G. Structure Theorems for Group Ring Codes with an Application to Self-Dual Codes. Designs, Codes and Cryptography 24, 5–14 (2001). https://doi.org/10.1023/A:1011299010894
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DOI: https://doi.org/10.1023/A:1011299010894