Y. Cengellenmis, A. Dertli, S. Dougherty, Adrian Korban, S. Șahinkaya, Deniz Ustun
In this paper, we show that one can construct a begin{document}$ G $end{document}-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which begin{document}$ G $end{document}-codes are reversible of index begin{document}$ alpha $end{document}. Additionally, we introduce a new family of rings, begin{document}$ {mathcal{F}}_{j,k} $end{document}, whose base is the finite field of order begin{document}$ 4 $end{document} and study reversible begin{document}$ G $end{document}-codes over this family of rings. Moreover, we present some possible applications of reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathcal{F}}_{j,k} $end{document} to reversible DNA codes. We construct many reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathbb{F}}_4 $end{document} of which some are optimal. These codes can be used to obtain reversible DNA codes.
In this paper, we show that one can construct a begin{document}$ G $end{document}-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which begin{document}$ G $end{document}-codes are reversible of index begin{document}$ alpha $end{document}. Additionally, we introduce a new family of rings, begin{document}$ {mathcal{F}}_{j,k} $end{document}, whose base is the finite field of order begin{document}$ 4 $end{document} and study reversible begin{document}$ G $end{document}-codes over this family of rings. Moreover, we present some possible applications of reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathcal{F}}_{j,k} $end{document} to reversible DNA codes. We construct many reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathbb{F}}_4 $end{document} of which some are optimal. These codes can be used to obtain reversible DNA codes.
{"title":"Reversible $ G $-codes over the ring $ {mathcal{F}}_{j,k} $ with applications to DNA codes","authors":"Y. Cengellenmis, A. Dertli, S. Dougherty, Adrian Korban, S. Șahinkaya, Deniz Ustun","doi":"10.3934/amc.2021056","DOIUrl":"https://doi.org/10.3934/amc.2021056","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we show that one can construct a <inline-formula><tex-math id=\"M3\">begin{document}$ G $end{document}</tex-math></inline-formula>-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which <inline-formula><tex-math id=\"M4\">begin{document}$ G $end{document}</tex-math></inline-formula>-codes are reversible of index <inline-formula><tex-math id=\"M5\">begin{document}$ alpha $end{document}</tex-math></inline-formula>. Additionally, we introduce a new family of rings, <inline-formula><tex-math id=\"M6\">begin{document}$ {mathcal{F}}_{j,k} $end{document}</tex-math></inline-formula>, whose base is the finite field of order <inline-formula><tex-math id=\"M7\">begin{document}$ 4 $end{document}</tex-math></inline-formula> and study reversible <inline-formula><tex-math id=\"M8\">begin{document}$ G $end{document}</tex-math></inline-formula>-codes over this family of rings. Moreover, we present some possible applications of reversible <inline-formula><tex-math id=\"M9\">begin{document}$ G $end{document}</tex-math></inline-formula>-codes over <inline-formula><tex-math id=\"M10\">begin{document}$ {mathcal{F}}_{j,k} $end{document}</tex-math></inline-formula> to reversible DNA codes. We construct many reversible <inline-formula><tex-math id=\"M11\">begin{document}$ G $end{document}</tex-math></inline-formula>-codes over <inline-formula><tex-math id=\"M12\">begin{document}$ {mathbb{F}}_4 $end{document}</tex-math></inline-formula> of which some are optimal. These codes can be used to obtain reversible DNA codes.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"233 1","pages":"1406-1421"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89150752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dualities for codes over finite Abelian groups","authors":"S. Dougherty","doi":"10.3934/amc.2023023","DOIUrl":"https://doi.org/10.3934/amc.2023023","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"18 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73590351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let begin{document}$ mathbb{Z}_4 $end{document} be the ring of integers modulo begin{document}$ 4 $end{document}. This paper studies mixed alphabets begin{document}$ mathbb{Z}_4mathbb{Z}_4[u^3] $end{document}-additive cyclic and begin{document}$ lambda $end{document}-constacyclic codes for units begin{document}$ lambda = 1+2u^2,3+2u^2 $end{document}. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain begin{document}$ mathbb{Z}_4 $end{document}-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.
Let begin{document}$ mathbb{Z}_4 $end{document} be the ring of integers modulo begin{document}$ 4 $end{document}. This paper studies mixed alphabets begin{document}$ mathbb{Z}_4mathbb{Z}_4[u^3] $end{document}-additive cyclic and begin{document}$ lambda $end{document}-constacyclic codes for units begin{document}$ lambda = 1+2u^2,3+2u^2 $end{document}. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain begin{document}$ mathbb{Z}_4 $end{document}-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.
{"title":"On $ mathbb{Z}_4mathbb{Z}_4[u^3] $-additive constacyclic codes","authors":"O. Prakash, S. Yadav, H. Islam, P. Solé","doi":"10.3934/amc.2022017","DOIUrl":"https://doi.org/10.3934/amc.2022017","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M2\">begin{document}$ mathbb{Z}_4 $end{document}</tex-math></inline-formula> be the ring of integers modulo <inline-formula><tex-math id=\"M3\">begin{document}$ 4 $end{document}</tex-math></inline-formula>. This paper studies mixed alphabets <inline-formula><tex-math id=\"M4\">begin{document}$ mathbb{Z}_4mathbb{Z}_4[u^3] $end{document}</tex-math></inline-formula>-additive cyclic and <inline-formula><tex-math id=\"M5\">begin{document}$ lambda $end{document}</tex-math></inline-formula>-constacyclic codes for units <inline-formula><tex-math id=\"M6\">begin{document}$ lambda = 1+2u^2,3+2u^2 $end{document}</tex-math></inline-formula>. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain <inline-formula><tex-math id=\"M7\">begin{document}$ mathbb{Z}_4 $end{document}</tex-math></inline-formula>-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"13 1","pages":"246-261"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77720673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of begin{document}$ b $end{document} for a given begin{document}$ a $end{document} for which an begin{document}$ (a,b) $end{document} -absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of begin{document}$ b $end{document} for a given begin{document}$ a $end{document} for which an begin{document}$ (a,b) $end{document} -absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.
{"title":"Extremal absorbing sets in low-density parity-check codes","authors":"Emily McMillon, Allison Beemer, C. Kelley","doi":"10.3934/AMC.2021003","DOIUrl":"https://doi.org/10.3934/AMC.2021003","url":null,"abstract":"Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of begin{document}$ b $end{document} for a given begin{document}$ a $end{document} for which an begin{document}$ (a,b) $end{document} -absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"12 1","pages":"465-483"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76129946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva
The $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive codes are subgroups of $ mathbb{Z}_2^{alpha_1} times mathbb{Z}_4^{alpha_2} times mathbb{Z}_8^{alpha_3} $. A $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard code is a Hadamard code, which is the Gray map image of a $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive code. In this paper, we generalize some known results for $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard codes to $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes with $ alpha_1 neq 0 $, $ alpha_2 neq 0 $, and $ alpha_3 neq 0 $. First, we give a recursive construction of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive Hadamard codes of type $ (alpha_1, alpha_2, alpha_3;t_1, t_2, t_3) $ with $ t_1geq 1 $, $ t_2 geq 0 $, and $ t_3geq 1 $. It is known that each $ mathbb{Z}_4 $-linear Hadamard code is equivalent to a $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard code with $ alpha_1neq 0 $ and $ alpha_2neq 0 $. Unlike $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard codes, in general, this family of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes does not include the family of $ mathbb{Z}_4 $-linear or $ mathbb{Z}_8 $-linear Hadamard codes. We show that, for example, for length $ 2^{11} $, the constructed nonlinear $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes are not equivalent to each other, nor to any $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard, nor to any previously constructed $ mathbb{Z}_{2^s} $-Hadamard code, with $ sgeq 2 $. Finally, we also present other recursive constructions of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones.
{"title":"On recursive constructions of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes","authors":"Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva","doi":"10.3934/amc.2023047","DOIUrl":"https://doi.org/10.3934/amc.2023047","url":null,"abstract":"The $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive codes are subgroups of $ mathbb{Z}_2^{alpha_1} times mathbb{Z}_4^{alpha_2} times mathbb{Z}_8^{alpha_3} $. A $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard code is a Hadamard code, which is the Gray map image of a $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive code. In this paper, we generalize some known results for $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard codes to $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes with $ alpha_1 neq 0 $, $ alpha_2 neq 0 $, and $ alpha_3 neq 0 $. First, we give a recursive construction of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive Hadamard codes of type $ (alpha_1, alpha_2, alpha_3;t_1, t_2, t_3) $ with $ t_1geq 1 $, $ t_2 geq 0 $, and $ t_3geq 1 $. It is known that each $ mathbb{Z}_4 $-linear Hadamard code is equivalent to a $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard code with $ alpha_1neq 0 $ and $ alpha_2neq 0 $. Unlike $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard codes, in general, this family of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes does not include the family of $ mathbb{Z}_4 $-linear or $ mathbb{Z}_8 $-linear Hadamard codes. We show that, for example, for length $ 2^{11} $, the constructed nonlinear $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes are not equivalent to each other, nor to any $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard, nor to any previously constructed $ mathbb{Z}_{2^s} $-Hadamard code, with $ sgeq 2 $. Finally, we also present other recursive constructions of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135560703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is then used to determine the number of minimum weight codewords of a projective Reed-Muller code. Various formulas for the dimension of a projective Reed-Muller code, and their equivalences are also discussed.
{"title":"On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes","authors":"Sudhir R. Ghorpade, Rati Ludhani","doi":"10.3934/amc.2023035","DOIUrl":"https://doi.org/10.3934/amc.2023035","url":null,"abstract":"We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is then used to determine the number of minimum weight codewords of a projective Reed-Muller code. Various formulas for the dimension of a projective Reed-Muller code, and their equivalences are also discussed.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135699005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of it's dual is the MacWilliams Identity, first developed for the Hamming metric. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via (generalised) Krawtchouk polynomials. The functional transformation form can in particular be used to derive important moment identities for the weight distribution of codes. In this paper, we focus on codes in the skew rank metric. In these codes, the codewords are skew-symmetric matrices, and the distance between two matrices is the skew rank metric, which is half the rank of their difference. This paper develops a $ q $-analog MacWilliams Identity in the form of a functional transformation for codes based on skew-symmetric matrices under their associated skew rank metric. The method introduces a skew-$ q $ algebra and uses generalised Krawtchouk polynomials. Based on this new MacWilliams Identity, we then derive several moments of the skew rank distribution for these codes.
{"title":"The MacWilliams identity for the skew rank metric","authors":"Izzy Friedlander, Thanasis Bouganis, Maximilien Gadouleau","doi":"10.3934/amc.2023045","DOIUrl":"https://doi.org/10.3934/amc.2023045","url":null,"abstract":"The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of it's dual is the MacWilliams Identity, first developed for the Hamming metric. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via (generalised) Krawtchouk polynomials. The functional transformation form can in particular be used to derive important moment identities for the weight distribution of codes. In this paper, we focus on codes in the skew rank metric. In these codes, the codewords are skew-symmetric matrices, and the distance between two matrices is the skew rank metric, which is half the rank of their difference. This paper develops a $ q $-analog MacWilliams Identity in the form of a functional transformation for codes based on skew-symmetric matrices under their associated skew rank metric. The method introduces a skew-$ q $ algebra and uses generalised Krawtchouk polynomials. Based on this new MacWilliams Identity, we then derive several moments of the skew rank distribution for these codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135505953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a polynomial representation for additive cyclic codes over $ mathbb{F}_{p^2} $. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over $ mathbb{F}_p $. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over $ mathbb{F}_{p^2} $. Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over $ mathbb{F}_{4} $.
{"title":"Polynomial representation of additive cyclic codes and new quantum codes","authors":"Reza Dastbasteh, Khalil Shivji","doi":"10.3934/amc.2023036","DOIUrl":"https://doi.org/10.3934/amc.2023036","url":null,"abstract":"We give a polynomial representation for additive cyclic codes over $ mathbb{F}_{p^2} $. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over $ mathbb{F}_p $. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over $ mathbb{F}_{p^2} $. Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over $ mathbb{F}_{4} $.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136257632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present a method for constructing extremal $ mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ kinleft{9,10,12,13,14,15,16right} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ kinleft{7,9,10,12,13,14,15,16right} $. For the length 40, constructed extremal $ mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ kinleft{7,10,11,15,16right} $. There are at least 40 new Type Ⅱ extremal $ mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ mathbb{Z}_{4} $-codes.
{"title":"Construction of extremal $ mathbb{Z}_{4} $-codes using a neighborhood search algorithm","authors":"Dean Crnković, Matteo Mravić, Sanja Rukavina","doi":"10.3934/amc.2023039","DOIUrl":"https://doi.org/10.3934/amc.2023039","url":null,"abstract":"In this paper, we present a method for constructing extremal $ mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ kinleft{9,10,12,13,14,15,16right} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ kinleft{7,9,10,12,13,14,15,16right} $. For the length 40, constructed extremal $ mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ kinleft{7,10,11,15,16right} $. There are at least 40 new Type Ⅱ extremal $ mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ mathbb{Z}_{4} $-codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136366914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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