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Some results on BCH codes of length $ frac{{{q^m} + 1}}{2} $ 长度为$ frac{{{q^m} + 1}}{2} $的BCH码的一些结果
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023010
Jiayuan Zhang, Ping Li, X. Kai, Zhonghua Sun
{"title":"Some results on BCH codes of length $ frac{{{q^m} + 1}}{2} $","authors":"Jiayuan Zhang, Ping Li, X. Kai, Zhonghua Sun","doi":"10.3934/amc.2023010","DOIUrl":"https://doi.org/10.3934/amc.2023010","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"62 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91119323","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}
引用次数: 1
The error-correcting pair for direct sum codes 直接和码的纠错对
4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023046
Boyi He, Qunying Liao
{"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}
引用次数: 0
The weight distributions of several classes of few-weight linear codes 几种少权线性码的权值分布
4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023037
Huan Sun, Qin Yue, Xue Jia
{"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}
引用次数: 0
Efficient keyword search on encrypted dynamic cloud data 对加密动态云数据的高效关键字搜索
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2022101
Laltu Sardar, Binanda Sengupta, S. Ruj
{"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}
引用次数: 0
Two pointsets in $ mathrm{PG}(2,q^n) $ and the associated codes $ mathm {PG}(2,q^n) $中的两个点集及其相关代码
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2022006
V. Napolitano, O. Polverino, Paolo Santonastaso, Ferdinando Zullo
{"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}
引用次数: 2
Parameters of some BCH codes over $ mathbb{F}_q $ of length $ frac{q^m-1}{2} $ 长度为$ frac{q^m-1}{2} $的$ mathbb{F}_q $上的一些BCH码的参数
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023007
Liqi Wang, Di Lu, Shixin Zhu
{"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}
引用次数: 0
Nonexistence of some four dimensional linear codes attaining the Griesmer bound 某些达到Griesmer界的四维线性码的不存在性
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023024
W. Ma, Jinquan Luo
{"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}
引用次数: 0
Fractional non-norm elements for division algebras, and an application to Cyclic Learning with Errors 除法代数的分数阶非范数元及其在带误差循环学习中的应用
4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023043
Andrew Mendelsohn, Cong Ling
{"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}
引用次数: 0
Asymptotically good generalized quasi-cyclic codes over finite chain rings 有限链环上渐近良好的广义拟循环码
4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023034
Xiangrui Meng, Jian Gao, Fang-Wei Fu
{"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}
引用次数: 0
Dualities for codes over finite Abelian groups 有限阿贝尔群上码的对偶性
IF 0.9 4区 计算机科学
Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI: 10.3934/amc.2023023
S. Dougherty
{"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}
引用次数: 1
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