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Korban Adrian - One of the best experts on this subject based on the ideXlab platform.
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Group Codes, Composite Group Codes and Constructions of Self-Dual Codes
University of Chester, 2021Co-Authors: Korban AdrianAbstract:The main research presented in this thesis is around constructing binary self-dual codes using group Rings together with some well-known code construction methods and the study of group codes and composite group codes over different alphabets. Both these families of codes are generated by the elements that come from group Rings. A search for binary self-dual codes with new weight enumerators is an ongoing research area in algebraic coding theory. For this reason, we present a generator matrix in which we employ the idea of a bisymmetric matrix with its entries being the block matrices that come from group Rings and give the necessary conditions for this generator matrix to produce a self-dual code over a fi nite commutative Frobenius Ring. Together with our generator matrix and some well-known code construction methods, we find many binary self-dual codes with parameters [68, 34, 12] that have weight enumerators that were not known in the literature before. There is an extensive literature on the study of different families of codes over different alphabets and speci fically finite fi elds and finite commutative Rings. The study of codes over Rings opens up a new direction for constructing new binary self-dual codes with a rich automorphism group via the algebraic structure of the Rings through the Gray maps associated with them. In this thesis, we introduce a new family of Rings, study its algebraic structure and show that each member of this family is a commutative Frobenius Ring. Moreover, we study group codes over this new family of Rings and show that one can obtain codes with a rich automorphism group via the associated Gray map. We extend a well established isomorphism between group Rings and the subRing of the n x n matrices and show its applications to algebraic coding theory. Our extension enables one to construct many complex n x n matrices over the Ring R that are fully de ned by the elements appeaRing in the first row. This property allows one to build generator matrices with these complex matrices so that the search field is practical in terms of the computational times. We show how these complex matrices are constructed using group Rings, study their properties and present many interesting examples of complex matrices over the Ring R. Using our extended isomorphism, we de ne a new family of codes which we call the composite group codes or for simplicity, composite G-codes. We show that these new codes are ideals in the group Ring RG and prove that the dual of a composite G-code is also a composite G-code. Moreover, we study generator matrices of the form [In | Ω(v)]; where In is the n x n identity matrix and Ω(v) is the composite matrix that comes from the extended isomorphism mentioned earlier. In particular, we show when such generator matrices produce self-dual codes over finite commutative Frobenius Rings. Additionally, together with some generator matrices of the type [In | Ω(v)] and the well-known extension and neighbour methods, we fi nd many new binary self-dual codes with parameters [68, 34, 12]. Lastly in this work, we study composite G-codes over formal power series Rings and finite chain Rings. We extend many known results on projections and lifts of codes over these alphabets. We also extend some known results on γadic codes over the infi nite Ring R
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New singly and doubly even binary [72,36,12] self-dual codes from M 2(R)G - group matrix Rings
'Elsevier BV', 2021Co-Authors: Korban Adrian, Şahinkaya Serap, Ustun DenizAbstract:From Elsevier via Jisc Publications RouterHistory: accepted 2021-08-26, epub 2021-09-17, issued 2021-12-31Article version: AMPublication status: PublishedIn this work, we present a number of generator matrices of the form [ I 2 n | τ 2 ( v ) ] , where I 2 n is the 2 n × 2 n identity matrix, v is an element in the group matrix Ring M 2 ( R ) G and where R is a finite commutative Frobenius Ring and G is a finite group of order 18. We employ these generator matrices and search for binary [ 72 , 36 , 12 ] self-dual codes directly over the finite field F 2 . As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings
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Composite matrices from group Rings, composite G -codes and constructions of self-dual codes
Springer US, 2021Co-Authors: Dougherty, Steven T., Korban Adrian, Gildea Joe, Kaya AbidinAbstract:From SpRinger Nature via Jisc Publications RouterHistory: received 2020-02-25, rev-recd 2021-02-09, accepted 2021-04-17, registration 2021-04-17, pub-electronic 2021-05-19, online 2021-05-19, pub-print 2021-07Publication status: PublishedAbstract: In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters γ=7, 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructio
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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
'Springer Science and Business Media LLC', 2021Co-Authors: Dougherty Steven, Korban Adrian, Gildea Joe, Kaya AbidinAbstract:This document is the Accepted Manuscript version of a published work that appeared in final form in Designs, Codes and Cryptography. To access the final edited and published work see http://dx.doi.org/10.1007/s10623-021-00882-8In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions
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G -codes, self-dual G -codes and reversible G -codes over the Ring B j, k
Springer US, 2021Co-Authors: Dougherty S. T., Korban Adrian, Gildea Joe, Şahinkaya SerapAbstract:From SpRinger Nature via Jisc Publications RouterHistory: received 2020-09-25, accepted 2021-03-24, registration 2021-03-25, online 2021-05-03, pub-electronic 2021-05-03, pub-print 2021-09Publication status: PublishedAbstract: In this work, we study a new family of Rings, Bj, k, whose base field is the finite field Fpr. We study the structure of this family of Rings and show that each member of the family is a commutative Frobenius Ring. We define a Gray map for the new family of Rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj, k to a code over Bl, m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the Rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s
Gildea Joe - One of the best experts on this subject based on the ideXlab platform.
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Composite matrices from group Rings, composite G -codes and constructions of self-dual codes
Springer US, 2021Co-Authors: Dougherty, Steven T., Korban Adrian, Gildea Joe, Kaya AbidinAbstract:From SpRinger Nature via Jisc Publications RouterHistory: received 2020-02-25, rev-recd 2021-02-09, accepted 2021-04-17, registration 2021-04-17, pub-electronic 2021-05-19, online 2021-05-19, pub-print 2021-07Publication status: PublishedAbstract: In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters γ=7, 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructio
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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
'Springer Science and Business Media LLC', 2021Co-Authors: Dougherty Steven, Korban Adrian, Gildea Joe, Kaya AbidinAbstract:This document is the Accepted Manuscript version of a published work that appeared in final form in Designs, Codes and Cryptography. To access the final edited and published work see http://dx.doi.org/10.1007/s10623-021-00882-8In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions
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G -codes, self-dual G -codes and reversible G -codes over the Ring B j, k
Springer US, 2021Co-Authors: Dougherty S. T., Korban Adrian, Gildea Joe, Şahinkaya SerapAbstract:From SpRinger Nature via Jisc Publications RouterHistory: received 2020-09-25, accepted 2021-03-24, registration 2021-03-25, online 2021-05-03, pub-electronic 2021-05-03, pub-print 2021-09Publication status: PublishedAbstract: In this work, we study a new family of Rings, Bj, k, whose base field is the finite field Fpr. We study the structure of this family of Rings and show that each member of the family is a commutative Frobenius Ring. We define a Gray map for the new family of Rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj, k to a code over Bl, m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the Rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s
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G-Codes, self-dual G-Codes and reversible G-Codes over the Ring Bj,k
'Springer Science and Business Media LLC', 2021Co-Authors: Dougherty Steven, Korban Adrian, Gildea Joe, Şahinkaya SerapAbstract:This version of the article has been accepted for publication, after peer review (when applicable) and is subject to SpRinger Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s12095-021-00487-xIn this work, we study a new family of Rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of Rings and show that each member of the family is a commutative Frobenius Ring. We define a Gray map for the new family of Rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj,k to a code over Bl,m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the Rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s
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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
2020Co-Authors: Dougherty, Steven T., Korban Adrian, Gildea Joe, Kaya AbidinAbstract:In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We define quasi-composite G-codes and give a construction of these codes. We also study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma=7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.Comment: 33 page
Kaya Abidin - One of the best experts on this subject based on the ideXlab platform.
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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
'Springer Science and Business Media LLC', 2021Co-Authors: Dougherty Steven, Korban Adrian, Gildea Joe, Kaya AbidinAbstract:This document is the Accepted Manuscript version of a published work that appeared in final form in Designs, Codes and Cryptography. To access the final edited and published work see http://dx.doi.org/10.1007/s10623-021-00882-8In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions
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Composite matrices from group Rings, composite G -codes and constructions of self-dual codes
Springer US, 2021Co-Authors: Dougherty, Steven T., Korban Adrian, Gildea Joe, Kaya AbidinAbstract:From SpRinger Nature via Jisc Publications RouterHistory: received 2020-02-25, rev-recd 2021-02-09, accepted 2021-04-17, registration 2021-04-17, pub-electronic 2021-05-19, online 2021-05-19, pub-print 2021-07Publication status: PublishedAbstract: In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters γ=7, 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructio
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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
2020Co-Authors: Dougherty, Steven T., Korban Adrian, Gildea Joe, Kaya AbidinAbstract:In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We define quasi-composite G-codes and give a construction of these codes. We also study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma=7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.Comment: 33 page
Dougherty, Steven T. - One of the best experts on this subject based on the ideXlab platform.
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Composite matrices from group Rings, composite G -codes and constructions of self-dual codes
Springer US, 2021Co-Authors: Dougherty, Steven T., Korban Adrian, Gildea Joe, Kaya AbidinAbstract:From SpRinger Nature via Jisc Publications RouterHistory: received 2020-02-25, rev-recd 2021-02-09, accepted 2021-04-17, registration 2021-04-17, pub-electronic 2021-05-19, online 2021-05-19, pub-print 2021-07Publication status: PublishedAbstract: In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters γ=7, 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructio
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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
2020Co-Authors: Dougherty, Steven T., Korban Adrian, Gildea Joe, Kaya AbidinAbstract:In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We define quasi-composite G-codes and give a construction of these codes. We also study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma=7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.Comment: 33 page
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Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings
2016Co-Authors: Dougherty, Steven T., Taylor Rhian, Gildea Joe, Tylyshchak AlexanderAbstract:We give constructions of self-dual and formally self-dual codes from group Rings where the Ring is a finite commutative Frobenius Ring. We improve the existing construction given in \cite{Hurley1} 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 correspond to ideals in the group Ring $RG$ and as such 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
Dougherty Steven - One of the best experts on this subject based on the ideXlab platform.
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G-Codes, self-dual G-Codes and reversible G-Codes over the Ring Bj,k
'Springer Science and Business Media LLC', 2021Co-Authors: Dougherty Steven, Korban Adrian, Gildea Joe, Şahinkaya SerapAbstract:This version of the article has been accepted for publication, after peer review (when applicable) and is subject to SpRinger Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s12095-021-00487-xIn this work, we study a new family of Rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of Rings and show that each member of the family is a commutative Frobenius Ring. We define a Gray map for the new family of Rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj,k to a code over Bl,m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the Rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s
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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
'Springer Science and Business Media LLC', 2021Co-Authors: Dougherty Steven, Korban Adrian, Gildea Joe, Kaya AbidinAbstract:This document is the Accepted Manuscript version of a published work that appeared in final form in Designs, Codes and Cryptography. To access the final edited and published work see http://dx.doi.org/10.1007/s10623-021-00882-8In this work, we define composite matrices which are derived from group Rings. We extend the idea of G-codes to composite G-codes. We show that these codes 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 composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions
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Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes
'Springer Science and Business Media LLC', 2017Co-Authors: Dougherty Steven, Taylor Rhian, Gildea Joe, Tylyshchak AlexanderAbstract:The final publication is available at SpRinger via http://dx.doi.org/10.1007/s10623-017-0440-7We 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
