Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes
dc.contributor.author | Dougherty, Steven | * |
dc.contributor.author | Gildea, Joe | * |
dc.contributor.author | Taylor, Rhian | * |
dc.contributor.author | Tylyshchak, Alexander | * |
dc.date.accessioned | 2017-11-07T13:07:30Z | |
dc.date.available | 2017-11-07T13:07:30Z | |
dc.date.issued | 2017-11-15 | |
dc.identifier.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 | en |
dc.identifier.issn | 0925-1022 | |
dc.identifier.doi | 10.1007/s10623-017-0440-7 | |
dc.identifier.uri | http://hdl.handle.net/10034/620712 | |
dc.description | The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-017-0440-7 | |
dc.description.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. | |
dc.language.iso | en | en |
dc.publisher | Springer | en |
dc.relation.url | https://link.springer.com/article/10.1007/s10623-017-0440-7 | en |
dc.rights.uri | http://creativecommons.org/publicdomain/mark/1.0/ | en |
dc.subject | Mathematics | en |
dc.subject | Algebraic Coding Theory | en |
dc.title | Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes | en |
dc.type | Article | en |
dc.identifier.eissn | 1573-7586 | |
dc.contributor.department | University of Scranton; University of Chester; Uzhgorod State University | en |
dc.identifier.journal | Designs, Codes and Cryptography | |
dc.date.accepted | 2017-11-06 | |
or.grant.openaccess | Yes | en |
rioxxterms.funder | University of Chester | en |
rioxxterms.identifier.project | Internally funded - Mathematics Department - 2015/16 | en |
rioxxterms.version | AM | en |
rioxxterms.versionofrecord | https://doi.org/10.1007/s10623-017-0440-7 | |
rioxxterms.licenseref.startdate | 2018-11-15 | |
html.description.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. | |
rioxxterms.publicationdate | 2017-11-15 |