The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. In this paper, we focus our study on the error-correcting pair for the direct sum code of two linear codes with an error-correcting pair. Firstly, for the direct sum code $ mathcal{C} $ of two linear codes with an error-correcting pair, several sufficient conditions for $ mathcal{C} $ with an error-correcting pair are given. Secondly, for the direct sum code $ mathcal{C} $ of two Maximal Distance Separable linear codes, two Near-Maximal Distance Separable linear codes, or a Maximal Distance Separable linear code and a Near-Maximal Distance Separable linear code, several sufficient conditions for $ mathcal{C} $ with an error-correcting pair are given, respectively. And then, we introduce the corresponding decoding procedure of the direct sum code with an error-correcting pair, and give several examples.
{"title":"The error-correcting pair for direct sum codes","authors":"Boyi He, Qunying Liao","doi":"10.3934/amc.2023046","DOIUrl":"https://doi.org/10.3934/amc.2023046","url":null,"abstract":"The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. In this paper, we focus our study on the error-correcting pair for the direct sum code of two linear codes with an error-correcting pair. Firstly, for the direct sum code $ mathcal{C} $ of two linear codes with an error-correcting pair, several sufficient conditions for $ mathcal{C} $ with an error-correcting pair are given. Secondly, for the direct sum code $ mathcal{C} $ of two Maximal Distance Separable linear codes, two Near-Maximal Distance Separable linear codes, or a Maximal Distance Separable linear code and a Near-Maximal Distance Separable linear code, several sufficient conditions for $ mathcal{C} $ with an error-correcting pair are given, respectively. And then, we introduce the corresponding decoding procedure of the direct sum code with an error-correcting pair, and give several examples.","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":"135505507","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 data storage systems, authentication codes, association schemes, and some other fields, linear codes with few weights play an important role. In this paper, we construct six classes of few weights linear codes over $ mathbb F_{q} $ and use Gauss periods in the semi-primitive case to determine their weight distributions. There are some linear codes with few weights whose duals are almost Maximum Distance Separable (AMDS) codes and some Maximum Distance Separable (MDS) codes.
{"title":"The weight distributions of several classes of few-weight linear codes","authors":"Huan Sun, Qin Yue, Xue Jia","doi":"10.3934/amc.2023037","DOIUrl":"https://doi.org/10.3934/amc.2023037","url":null,"abstract":"In data storage systems, authentication codes, association schemes, and some other fields, linear codes with few weights play an important role. In this paper, we construct six classes of few weights linear codes over $ mathbb F_{q} $ and use Gauss periods in the semi-primitive case to determine their weight distributions. There are some linear codes with few weights whose duals are almost Maximum Distance Separable (AMDS) codes and some Maximum Distance Separable (MDS) codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"14 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":"136258886","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":"Efficient keyword search on encrypted dynamic cloud data","authors":"Laltu Sardar, Binanda Sengupta, S. Ruj","doi":"10.3934/amc.2022101","DOIUrl":"https://doi.org/10.3934/amc.2022101","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"23 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76036087","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}
V. Napolitano, O. Polverino, Paolo Santonastaso, Ferdinando Zullo
In this paper we consider two pointsets in begin{document}$ mathrm{PG}(2,q^n) $end{document} arising from a linear set begin{document}$ L $end{document} of rank begin{document}$ n $end{document} contained in a line of begin{document}$ mathrm{PG}(2,q^n) $end{document}: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set begin{document}$ L $end{document}. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing begin{document}$ L $end{document} to be an begin{document}$ {mathbb F}_{q} $end{document}-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the begin{document}$ Gammamathrm{L} $end{document}-class of begin{document}$ L $end{document} and the number of inequivalent codes we can construct starting from it.
In this paper we consider two pointsets in begin{document}$ mathrm{PG}(2,q^n) $end{document} arising from a linear set begin{document}$ L $end{document} of rank begin{document}$ n $end{document} contained in a line of begin{document}$ mathrm{PG}(2,q^n) $end{document}: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set begin{document}$ L $end{document}. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing begin{document}$ L $end{document} to be an begin{document}$ {mathbb F}_{q} $end{document}-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the begin{document}$ Gammamathrm{L} $end{document}-class of begin{document}$ L $end{document} and the number of inequivalent codes we can construct starting from it.
{"title":"Two pointsets in $ mathrm{PG}(2,q^n) $ and the associated codes","authors":"V. Napolitano, O. Polverino, Paolo Santonastaso, Ferdinando Zullo","doi":"10.3934/amc.2022006","DOIUrl":"https://doi.org/10.3934/amc.2022006","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we consider two pointsets in <inline-formula><tex-math id=\"M2\">begin{document}$ mathrm{PG}(2,q^n) $end{document}</tex-math></inline-formula> arising from a linear set <inline-formula><tex-math id=\"M3\">begin{document}$ L $end{document}</tex-math></inline-formula> of rank <inline-formula><tex-math id=\"M4\">begin{document}$ n $end{document}</tex-math></inline-formula> contained in a line of <inline-formula><tex-math id=\"M5\">begin{document}$ mathrm{PG}(2,q^n) $end{document}</tex-math></inline-formula>: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set <inline-formula><tex-math id=\"M6\">begin{document}$ L $end{document}</tex-math></inline-formula>. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing <inline-formula><tex-math id=\"M7\">begin{document}$ L $end{document}</tex-math></inline-formula> to be an <inline-formula><tex-math id=\"M8\">begin{document}$ {mathbb F}_{q} $end{document}</tex-math></inline-formula>-linear set with a <i>short</i> weight distribution, then the associated codes have <i>few weights</i>. We conclude the paper by providing a connection between the <inline-formula><tex-math id=\"M9\">begin{document}$ Gammamathrm{L} $end{document}</tex-math></inline-formula>-class of <inline-formula><tex-math id=\"M10\">begin{document}$ L $end{document}</tex-math></inline-formula> and the number of inequivalent codes we can construct starting from it.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"114 1","pages":"227-245"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77722569","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":"Parameters of some BCH codes over $ mathbb{F}_q $ of length $ frac{q^m-1}{2} $","authors":"Liqi Wang, Di Lu, Shixin Zhu","doi":"10.3934/amc.2023007","DOIUrl":"https://doi.org/10.3934/amc.2023007","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"6 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73165496","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":"Nonexistence of some four dimensional linear codes attaining the Griesmer bound","authors":"W. Ma, Jinquan Luo","doi":"10.3934/amc.2023024","DOIUrl":"https://doi.org/10.3934/amc.2023024","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"17 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82638415","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}
Given a cyclotomic field $ K $ and a finite Galois extension $ L $, we discuss the construction of unit-magnitude elements in $ K $ which are not in the image of the field norm map $ N_{L/K}(L^times) $. We observe that the construction of Elia, Sethuraman, and Kumar extends to all cyclotomic fields whose rings of integers are a principal ideal domain, a fact we have not seen appear elsewhere in the literature. We then prove a number of lemmas concerning non-norm elements, and extend the above results to hold for arbitrary cyclotomic ground fields. We give examples of towers of fields and corresponding non-norm elements in both instances. Finally, we apply this to cryptography, defining a novel variant of Learning with Errors, defined over cyclic division algebras with fractional unit-magnitude non-norm elements, and reduce lattice problems defined over ideals in maximal orders in such algebras to the search problem for this form of LWE.
{"title":"Fractional non-norm elements for division algebras, and an application to Cyclic Learning with Errors","authors":"Andrew Mendelsohn, Cong Ling","doi":"10.3934/amc.2023043","DOIUrl":"https://doi.org/10.3934/amc.2023043","url":null,"abstract":"Given a cyclotomic field $ K $ and a finite Galois extension $ L $, we discuss the construction of unit-magnitude elements in $ K $ which are not in the image of the field norm map $ N_{L/K}(L^times) $. We observe that the construction of Elia, Sethuraman, and Kumar extends to all cyclotomic fields whose rings of integers are a principal ideal domain, a fact we have not seen appear elsewhere in the literature. We then prove a number of lemmas concerning non-norm elements, and extend the above results to hold for arbitrary cyclotomic ground fields. We give examples of towers of fields and corresponding non-norm elements in both instances. Finally, we apply this to cryptography, defining a novel variant of Learning with Errors, defined over cyclic division algebras with fractional unit-magnitude non-norm elements, and reduce lattice problems defined over ideals in maximal orders in such algebras to the search problem for this form of LWE.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"27 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":"135212418","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 construct a class of generalized quasi-cyclic (GQC) codes with index $ ell $ over finite chain rings. Based on probabilistic arguments, we discuss asymptotic rates and relative distances of this class of codes. As a result, we show that GQC codes with index $ ell $ over finite chain rings are asymptotically good.
在有限链环上构造了一类索引为$ well $的广义拟循环码。基于概率论证，讨论了这类码的渐近速率和相对距离。结果表明，有限链环上索引为$ well $的GQC码是渐近好的。
{"title":"Asymptotically good generalized quasi-cyclic codes over finite chain rings","authors":"Xiangrui Meng, Jian Gao, Fang-Wei Fu","doi":"10.3934/amc.2023034","DOIUrl":"https://doi.org/10.3934/amc.2023034","url":null,"abstract":"In this paper, we construct a class of generalized quasi-cyclic (GQC) codes with index $ ell $ over finite chain rings. Based on probabilistic arguments, we discuss asymptotic rates and relative distances of this class of codes. As a result, we show that GQC codes with index $ ell $ over finite chain rings are asymptotically good.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"109 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":"135699007","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}
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