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

Hdl Handle:
http://hdl.handle.net/10034/620712
Title:
Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes
Authors:
Dougherty, Steven; Gildea, Joe; Taylor, Rhian; Tylyschak, Alexander
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.
Affiliation:
University of Scranton; University of Chester; Uzhgorod State University
Citation:
Dougherty, S., Gildea, J., Taylor, R., & Tylyschak, A. (2017 - in press). Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes. Designs, Codes and Cryptography.
Publisher:
Springer
Journal:
Designs, Codes and Cryptography
Publication Date:
2017
URI:
http://hdl.handle.net/10034/620712
Additional Links:
https://link.springer.com/journal/10623
Type:
Article
Language:
en
EISSN:
1573-7586
Appears in Collections:
Mathematics

Full metadata record

DC FieldValue Language
dc.contributor.authorDougherty, Stevenen
dc.contributor.authorGildea, Joeen
dc.contributor.authorTaylor, Rhianen
dc.contributor.authorTylyschak, Alexanderen
dc.date.accessioned2017-11-07T13:07:30Z-
dc.date.available2017-11-07T13:07:30Z-
dc.date.issued2017-
dc.identifier.citationDougherty, S., Gildea, J., Taylor, R., & Tylyschak, A. (2017 - in press). Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes. Designs, Codes and Cryptography.en
dc.identifier.urihttp://hdl.handle.net/10034/620712-
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.en
dc.language.isoenen
dc.publisherSpringeren
dc.relation.urlhttps://link.springer.com/journal/10623en
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 Cryptographyen
dc.date.accepted2017-11-06-
or.grant.openaccessYesen
rioxxterms.funderUniversity of Chesteren
rioxxterms.identifier.projectInternally funded - Mathematics Department - 2015/16en
rioxxterms.versionAMen
rioxxterms.licenseref.startdate2217-11-07-
This item is licensed under a Creative Commons License
Creative Commons
All Items in ChesterRep are protected by copyright, with all rights reserved, unless otherwise indicated.