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Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes

Dougherty, Steven
; 
Gildea, Joe
; 
Taylor, Rhian
; 
Tylyshchak, Alexander
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University of Scranton; University of Chester; Uzhgorod State University
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2017-11-15
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Mathematics
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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 [13] 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.
Citation
Dougherty, S., Gildea, J., Taylor, R., & Tylyschak, A. (2018). Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes. Designs, Codes and Cryptography, 86(9), 2115-2138. https://doi.org/10.1007/s10623-017-0440-7
Publisher
Springer
Journal
Designs, Codes and Cryptography
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URI
http://hdl.handle.net/10034/620712
DOI
10.1007/s10623-017-0440-7
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Article
Language
en
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The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-017-0440-7
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ISSN
0925-1022
EISSN
1573-7586
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https://link.springer.com/article/10.1007/s10623-017-0440-7
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