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dc.contributor.authorDougherty, Steven*
dc.contributor.authorGildea, Joe*
dc.contributor.authorTaylor, Rhian*
dc.contributor.authorTylyshchak, Alexander*
dc.date.accessioned2017-11-07T13:07:30Z
dc.date.available2017-11-07T13:07:30Z
dc.date.issued2017-11-15
dc.identifier.citationDougherty, 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/s10623-017-0440-7en
dc.identifier.issn0925-1022
dc.identifier.doi10.1007/s10623-017-0440-7
dc.identifier.urihttp://hdl.handle.net/10034/620712
dc.descriptionThe final publication is available at Springer via http://dx.doi.org/10.1007/s10623-017-0440-7
dc.description.abstractWe 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.isoenen
dc.publisherSpringeren
dc.relation.urlhttps://link.springer.com/article/10.1007/s10623-017-0440-7en
dc.rights.urihttp://creativecommons.org/publicdomain/mark/1.0/en
dc.subjectMathematicsen
dc.subjectAlgebraic Coding Theoryen
dc.titleGroup Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codesen
dc.typeArticleen
dc.identifier.eissn1573-7586
dc.contributor.departmentUniversity of Scranton; University of Chester; Uzhgorod State Universityen
dc.identifier.journalDesigns, Codes and Cryptography
dc.date.accepted2017-11-06
or.grant.openaccessYesen
rioxxterms.funderUniversity of Chesteren
rioxxterms.identifier.projectInternally funded - Mathematics Department - 2015/16en
rioxxterms.versionAMen
rioxxterms.versionofrecordhttps://doi.org/10.1007/s10623-017-0440-7
rioxxterms.licenseref.startdate2018-11-15
html.description.abstractWe 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.publicationdate2017-11-15


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