For an odd prime begin{document}$ p $end{document} and positive integers begin{document}$ m $end{document} and begin{document}$ ell $end{document}, let begin{document}$ mathbb{F}_{p^m} $end{document} be the finite field with begin{document}$ p^{m} $end{document} elements and begin{document}$ R_{ell,m} = mathbb{F}_{p^m}[v_1,v_2,dots,v_{ell}]/langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}rangle_{1leq i, jleq ell} $end{document}. Thus begin{document}$ R_{ell,m} $end{document} is a finite commutative non-chain ring of order begin{document}$ p^{2^{ell} m} $end{document} with characteristic begin{document}$ p $end{document}. In this paper, we aim to construct quantum codes from skew constacyclic codes over begin{document}$ R_{ell,m} $end{document}. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.
For an odd prime begin{document}$ p $end{document} and positive integers begin{document}$ m $end{document} and begin{document}$ ell $end{document}, let begin{document}$ mathbb{F}_{p^m} $end{document} be the finite field with begin{document}$ p^{m} $end{document} elements and begin{document}$ R_{ell,m} = mathbb{F}_{p^m}[v_1,v_2,dots,v_{ell}]/langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}rangle_{1leq i, jleq ell} $end{document}. Thus begin{document}$ R_{ell,m} $end{document} is a finite commutative non-chain ring of order begin{document}$ p^{2^{ell} m} $end{document} with characteristic begin{document}$ p $end{document}. In this paper, we aim to construct quantum codes from skew constacyclic codes over begin{document}$ R_{ell,m} $end{document}. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.
{"title":"New quantum codes from skew constacyclic codes","authors":"Ram Krishna Verma, O. Prakash, A. Singh, H. Islam","doi":"10.3934/amc.2021028","DOIUrl":"https://doi.org/10.3934/amc.2021028","url":null,"abstract":"<p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id=\"M1\">begin{document}$ p $end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id=\"M2\">begin{document}$ m $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M3\">begin{document}$ ell $end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id=\"M4\">begin{document}$ mathbb{F}_{p^m} $end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id=\"M5\">begin{document}$ p^{m} $end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id=\"M6\">begin{document}$ R_{ell,m} = mathbb{F}_{p^m}[v_1,v_2,dots,v_{ell}]/langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}rangle_{1leq i, jleq ell} $end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id=\"M7\">begin{document}$ R_{ell,m} $end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id=\"M8\">begin{document}$ p^{2^{ell} m} $end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id=\"M9\">begin{document}$ p $end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id=\"M10\">begin{document}$ R_{ell,m} $end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"116 1","pages":"900-919"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90554973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the fundamental problems in coding theory is to find begin{document}$ n_q(k,d) $end{document}, the minimum length begin{document}$ n $end{document} for which a linear code of length begin{document}$ n $end{document}, dimension begin{document}$ k $end{document}, and the minimum weight begin{document}$ d $end{document} over the field of order begin{document}$ q $end{document} exists. The problem of determining the values of begin{document}$ n_q(k,d) $end{document} is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine begin{document}$ n_3(6,d) $end{document} for some values of begin{document}$ d $end{document} by proving the nonexistence of linear codes with certain parameters.
One of the fundamental problems in coding theory is to find begin{document}$ n_q(k,d) $end{document}, the minimum length begin{document}$ n $end{document} for which a linear code of length begin{document}$ n $end{document}, dimension begin{document}$ k $end{document}, and the minimum weight begin{document}$ d $end{document} over the field of order begin{document}$ q $end{document} exists. The problem of determining the values of begin{document}$ n_q(k,d) $end{document} is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine begin{document}$ n_3(6,d) $end{document} for some values of begin{document}$ d $end{document} by proving the nonexistence of linear codes with certain parameters.
{"title":"Nonexistence of some ternary linear codes with minimum weight -2 modulo 9","authors":"Toshiharu Sawashima, T. Maruta","doi":"10.3934/amc.2021052","DOIUrl":"https://doi.org/10.3934/amc.2021052","url":null,"abstract":"<p style='text-indent:20px;'>One of the fundamental problems in coding theory is to find <inline-formula><tex-math id=\"M3\">begin{document}$ n_q(k,d) $end{document}</tex-math></inline-formula>, the minimum length <inline-formula><tex-math id=\"M4\">begin{document}$ n $end{document}</tex-math></inline-formula> for which a linear code of length <inline-formula><tex-math id=\"M5\">begin{document}$ n $end{document}</tex-math></inline-formula>, dimension <inline-formula><tex-math id=\"M6\">begin{document}$ k $end{document}</tex-math></inline-formula>, and the minimum weight <inline-formula><tex-math id=\"M7\">begin{document}$ d $end{document}</tex-math></inline-formula> over the field of order <inline-formula><tex-math id=\"M8\">begin{document}$ q $end{document}</tex-math></inline-formula> exists. The problem of determining the values of <inline-formula><tex-math id=\"M9\">begin{document}$ n_q(k,d) $end{document}</tex-math></inline-formula> is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine <inline-formula><tex-math id=\"M10\">begin{document}$ n_3(6,d) $end{document}</tex-math></inline-formula> for some values of <inline-formula><tex-math id=\"M11\">begin{document}$ d $end{document}</tex-math></inline-formula> by proving the nonexistence of linear codes with certain parameters.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"77 1","pages":"1338-1357"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90358202","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}
Regev (2005) introduced the learning with errors (LWE) problem and showed a quantum reduction from a worst case lattice problem to LWE. Building on the work of Peikert (2009), a classical reduction from the gap shortest vector problem to LWE was obtained by Brakerski et al. (2013). A concrete security analysis of Regev's reduction by Chatterjee et al. (2016) identified a huge tightness gap. The present work performs a concrete analysis of the tightness gap in the classical reduction of Brakerski et al. It turns out that the tightness gap in the Brakerski et al. classical reduction is even larger than the tightness gap in the quantum reduction of Regev. This casts doubts on the implication of the reduction to security assurance of practical cryptosystems.
{"title":"Classical reduction of gap SVP to LWE: A concrete security analysis","authors":"P. Sarkar, Subhadip Singha","doi":"10.3934/AMC.2021004","DOIUrl":"https://doi.org/10.3934/AMC.2021004","url":null,"abstract":"Regev (2005) introduced the learning with errors (LWE) problem and showed a quantum reduction from a worst case lattice problem to LWE. Building on the work of Peikert (2009), a classical reduction from the gap shortest vector problem to LWE was obtained by Brakerski et al. (2013). A concrete security analysis of Regev's reduction by Chatterjee et al. (2016) identified a huge tightness gap. The present work performs a concrete analysis of the tightness gap in the classical reduction of Brakerski et al. It turns out that the tightness gap in the Brakerski et al. classical reduction is even larger than the tightness gap in the quantum reduction of Regev. This casts doubts on the implication of the reduction to security assurance of practical cryptosystems.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"34 1","pages":"484-499"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76826893","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":"Various structures of cyclic codes over the non-Frobenius ring $mathbb{F}_p[u, v] /leftlangle u^2, v^2, u v, v urightrangle$","authors":"H. Choi, Boran Kim","doi":"10.3934/amc.2023030","DOIUrl":"https://doi.org/10.3934/amc.2023030","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"33 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73135477","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":"PMNS for cryptography: A guided tour","authors":"Nicolas Méloni, François Palma, Pascal Véron","doi":"10.3934/amc.2023033","DOIUrl":"https://doi.org/10.3934/amc.2023033","url":null,"abstract":"","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":"135400791","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}
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}
{"title":"On the APN-ness and differential uniformity of some classes of $ (n,n) $-functions over $ mathbb{F}_2^n $","authors":"C. Carlet","doi":"10.3934/amc.2023027","DOIUrl":"https://doi.org/10.3934/amc.2023027","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"91 7 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87740709","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 propose a new class of optimal one-coincidence FHS (OC-FHS) sets with respect to the Peng-Fan bounds, including prime sequence sets and HMC sequence sets as special cases. Thereafter, through investigating their properties, we determine all of the FHS distances in the OC-FHS set. Finally, for a given positive integer, we also propose a new class of wide-gap one-coincidence FHS (WG-OC-FHS) sets where the FHS gap is larger than the given positive integer. Moreover, such a WG-OC-FHS set is optimal with respect to the WG-Lempel-Greenberger bound and the WG-Peng-Fan bounds simultaneously.
{"title":"A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets","authors":"Wenli Ren, Feng Wang","doi":"10.3934/AMC.2020131","DOIUrl":"https://doi.org/10.3934/AMC.2020131","url":null,"abstract":"In this paper, we propose a new class of optimal one-coincidence FHS (OC-FHS) sets with respect to the Peng-Fan bounds, including prime sequence sets and HMC sequence sets as special cases. Thereafter, through investigating their properties, we determine all of the FHS distances in the OC-FHS set. Finally, for a given positive integer, we also propose a new class of wide-gap one-coincidence FHS (WG-OC-FHS) sets where the FHS gap is larger than the given positive integer. Moreover, such a WG-OC-FHS set is optimal with respect to the WG-Lempel-Greenberger bound and the WG-Peng-Fan bounds simultaneously.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"9 1","pages":"342-352"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73884356","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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