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arXiv · 1604.07863

Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings

Steven T. Dougherty·Joe Gildea·Rhian Taylor·Alexander Tylyshchak
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Abstract

We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring $RG$ and as such must have an automorphism group that contains $G$ as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative $[72,36,16]$ Type~II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48.

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Steven T. Dougherty, Joe Gildea, Rhian Taylor, Alexander Tylyshchak. 2016-04-26. Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings. https://arxiv.org/abs/1604.07863

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