Group rings, G-codes and constructions of self-dual and formally self-dual codes
| Autor: | Steven T. Dougherty, Rhian Taylor, Joe Gildea, Alexander Tylyshchak |
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| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Finite group Ring (mathematics) Code (set theory) Applied Mathematics 020206 networking & telecommunications 0102 computer and information sciences 02 engineering and technology Coding theory 01 natural sciences Computer Science Applications Algebraic cycle 010201 computation theory & mathematics 0202 electrical engineering electronic engineering information engineering Commutative property Abstract algebra Group ring Mathematics |
| Zdroj: | Designs, Codes and Cryptography. 86:2115-2138 |
| ISSN: | 1573-7586 0925-1022 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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