Abstract
We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in Hurley (Int J Pure Appl Math 31(3):319–335, 2006) 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 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. We define quasi-G codes and give a construction of these codes.
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Notes
These groups are SmallGroup(24,i) for \(i \in \{3,6,8,10,12,13,14\}\) according to the GAP system [20].
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Communicated by C. Ding.
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Dougherty, S.T., Gildea, J., Taylor, R. et al. Group rings, G-codes and constructions of self-dual and formally self-dual codes. Des. Codes Cryptogr. 86, 2115–2138 (2018). https://doi.org/10.1007/s10623-017-0440-7
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DOI: https://doi.org/10.1007/s10623-017-0440-7