{"title":"Clay and Product-Matrix MSR Codes with Locality","authors":"Minhan Gao, Lukas Holzbaur, A. Wachter-Zeh","doi":"10.3934/amc.2023002","DOIUrl":"https://doi.org/10.3934/amc.2023002","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90561316","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 consider the quasi-twisted codes as contractions of quasi-cyclic codes and construct a family of $ q $-ary quasi-cyclic codes whose codewords have $ r $-divisible weights, where $ rmid q-1 $. We show that any quasi-cyclic code of co-index divisible by $ r $ is a direct sum of $ r $-divisible quasi-cyclic codes.
我们将拟扭曲码看作拟循环码的压缩，构造了一组$ q $一元的拟循环码，其码字具有$ r $-可整除的权值，其中$ r $中q $- 1 $。证明了任何可被r整除的拟循环码是r -可整除的拟循环码的直接和。
{"title":"Quasi-twisted codes as contractions of quasi-cyclic codes","authors":"Ferruh Özbudak, Buket Özkaya","doi":"10.3934/amc.2023041","DOIUrl":"https://doi.org/10.3934/amc.2023041","url":null,"abstract":"We consider the quasi-twisted codes as contractions of quasi-cyclic codes and construct a family of $ q $-ary quasi-cyclic codes whose codewords have $ r $-divisible weights, where $ rmid q-1 $. We show that any quasi-cyclic code of co-index divisible by $ r $ is a direct sum of $ r $-divisible quasi-cyclic codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010234","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}
Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal.
{"title":"Some subfield codes from MDS codes","authors":"Can Xiang, Jinquan Luo","doi":"10.3934/amc.2021023","DOIUrl":"https://doi.org/10.3934/amc.2021023","url":null,"abstract":"Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87980481","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}
Hengming Zhao, Rongcun Qin, M. Buratti, Dianhua Wu
{"title":"A few more optimal optical orthogonal codes with non-constant auto-correlation function","authors":"Hengming Zhao, Rongcun Qin, M. Buratti, Dianhua Wu","doi":"10.3934/amc.2023029","DOIUrl":"https://doi.org/10.3934/amc.2023029","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88100525","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":"The number of codes over rings of order $ 4 $ containing a hull of given type","authors":"Steven T. Dougherty, Esengül Saltürk","doi":"10.3934/amc.2023031","DOIUrl":"https://doi.org/10.3934/amc.2023031","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89790397","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":"Constructions of rotation symmetric Bent functions and Bent idempotent functions","authors":"Xiaoyan Chen, Sihong Su","doi":"10.3934/amc.2023022","DOIUrl":"https://doi.org/10.3934/amc.2023022","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77950218","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 study the begin{document}$ m $end{document} -ary sequences with (non-consecutive) two zero-symbols and at most two distinct autocorrelation coefficients, which are known as almost begin{document}$ m $end{document} -ary nearly perfect sequences. We show that these sequences are equivalent to begin{document}$ ell $end{document} -partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for begin{document}$ ell $end{document} -PDPDS. Finally, we present two construction methods for a family of almost quaternary sequences with at most two out-of-phase autocorrelation coefficients.
In this paper, we study the begin{document}$ m $end{document} -ary sequences with (non-consecutive) two zero-symbols and at most two distinct autocorrelation coefficients, which are known as almost begin{document}$ m $end{document} -ary nearly perfect sequences. We show that these sequences are equivalent to begin{document}$ ell $end{document} -partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for begin{document}$ ell $end{document} -PDPDS. Finally, we present two construction methods for a family of almost quaternary sequences with at most two out-of-phase autocorrelation coefficients.
{"title":"Partial direct product difference sets and almost quaternary sequences","authors":"Büsra Özden, Oğuz Yayla","doi":"10.3934/AMC.2021010","DOIUrl":"https://doi.org/10.3934/AMC.2021010","url":null,"abstract":"In this paper, we study the begin{document}$ m $end{document} -ary sequences with (non-consecutive) two zero-symbols and at most two distinct autocorrelation coefficients, which are known as almost begin{document}$ m $end{document} -ary nearly perfect sequences. We show that these sequences are equivalent to begin{document}$ ell $end{document} -partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for begin{document}$ ell $end{document} -PDPDS. Finally, we present two construction methods for a family of almost quaternary sequences with at most two out-of-phase autocorrelation coefficients.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78899369","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}
Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco
{"title":"New bounds for covering codes of radius 3 and codimension $ 3t+1 $","authors":"Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco","doi":"10.3934/amc.2023042","DOIUrl":"https://doi.org/10.3934/amc.2023042","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the number begin{document}$ R_n(t,N) $end{document} of tuplets begin{document}$ (x_1,ldots, x_n) $end{document} of congruence classes modulo begin{document}$ N $end{document} such that
begin{document}$ begin{equation*} x_1cdots x_n equiv t pmod{N}. end{equation*} $end{document}
As a result, we derive a recurrence for begin{document}$ R_n(t,N) $end{document} and prove some multiplicative properties of begin{document}$ R_n(t,N) $end{document}. Furthermore, we apply the result to study the probability distribution of Diffie-Hellman keys used in multiparty communication. We show that this probability distribution is not uniform.
We study the number begin{document}$ R_n(t,N) $end{document} of tuplets begin{document}$ (x_1,ldots, x_n) $end{document} of congruence classes modulo begin{document}$ N $end{document} such that begin{document}$ begin{equation*} x_1cdots x_n equiv t pmod{N}. end{equation*} $end{document} As a result, we derive a recurrence for begin{document}$ R_n(t,N) $end{document} and prove some multiplicative properties of begin{document}$ R_n(t,N) $end{document}. Furthermore, we apply the result to study the probability distribution of Diffie-Hellman keys used in multiparty communication. We show that this probability distribution is not uniform.
{"title":"On the number of factorizations of $ t $ mod $ N $ and the probability distribution of Diffie-Hellman secret keys for many users","authors":"A. Leibak","doi":"10.3934/amc.2021029","DOIUrl":"https://doi.org/10.3934/amc.2021029","url":null,"abstract":"<p style='text-indent:20px;'>We study the number <inline-formula><tex-math id=\"M3\">begin{document}$ R_n(t,N) $end{document}</tex-math></inline-formula> of tuplets <inline-formula><tex-math id=\"M4\">begin{document}$ (x_1,ldots, x_n) $end{document}</tex-math></inline-formula> of congruence classes modulo <inline-formula><tex-math id=\"M5\">begin{document}$ N $end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{equation*} x_1cdots x_n equiv t pmod{N}. end{equation*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>As a result, we derive a recurrence for <inline-formula><tex-math id=\"M6\">begin{document}$ R_n(t,N) $end{document}</tex-math></inline-formula> and prove some multiplicative properties of <inline-formula><tex-math id=\"M7\">begin{document}$ R_n(t,N) $end{document}</tex-math></inline-formula>. Furthermore, we apply the result to study the probability distribution of Diffie-Hellman keys used in multiparty communication. We show that this probability distribution is not uniform.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88190723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in begin{document}$ S_n $end{document}, the set of all permutations on begin{document}$ n $end{document} elements, under the Hamming metric. We prove the nonexistence of perfect begin{document}$ t $end{document}-error-correcting codes in begin{document}$ S_n $end{document} under the Hamming metric, for more values of begin{document}$ n $end{document} and begin{document}$ t $end{document}. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect begin{document}$ t $end{document}-error-correcting code in begin{document}$ S_n $end{document} under the Hamming metric for some begin{document}$ n $end{document} and begin{document}$ t = 1,2,3,4 $end{document}, or begin{document}$ 2t+1leq nleq max{4t^2e^{-2+1/t}-2,2t+1} $end{document} for begin{document}$ tgeq 2 $end{document}, or begin{document}$ min{frac{e}{2}sqrt{n+2},lfloorfrac{n-1}{2}rfloor}leq tleq lfloorfrac{n-1}{2}rfloor $end{document} for begin{document}$ ngeq 7 $end{document}, where begin{document}$ e $end{document} is the Napier's constant.
Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in begin{document}$ S_n $end{document}, the set of all permutations on begin{document}$ n $end{document} elements, under the Hamming metric. We prove the nonexistence of perfect begin{document}$ t $end{document}-error-correcting codes in begin{document}$ S_n $end{document} under the Hamming metric, for more values of begin{document}$ n $end{document} and begin{document}$ t $end{document}. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect begin{document}$ t $end{document}-error-correcting code in begin{document}$ S_n $end{document} under the Hamming metric for some begin{document}$ n $end{document} and begin{document}$ t = 1,2,3,4 $end{document}, or begin{document}$ 2t+1leq nleq max{4t^2e^{-2+1/t}-2,2t+1} $end{document} for begin{document}$ tgeq 2 $end{document}, or begin{document}$ min{frac{e}{2}sqrt{n+2},lfloorfrac{n-1}{2}rfloor}leq tleq lfloorfrac{n-1}{2}rfloor $end{document} for begin{document}$ ngeq 7 $end{document}, where begin{document}$ e $end{document} is the Napier's constant.
{"title":"New nonexistence results on perfect permutation codes under the hamming metric","authors":"Xiang Wang, Wenjuan Yin","doi":"10.3934/amc.2021058","DOIUrl":"https://doi.org/10.3934/amc.2021058","url":null,"abstract":"<p style='text-indent:20px;'>Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in <inline-formula><tex-math id=\"M1\">begin{document}$ S_n $end{document}</tex-math></inline-formula>, the set of all permutations on <inline-formula><tex-math id=\"M2\">begin{document}$ n $end{document}</tex-math></inline-formula> elements, under the Hamming metric. We prove the nonexistence of perfect <inline-formula><tex-math id=\"M3\">begin{document}$ t $end{document}</tex-math></inline-formula>-error-correcting codes in <inline-formula><tex-math id=\"M4\">begin{document}$ S_n $end{document}</tex-math></inline-formula> under the Hamming metric, for more values of <inline-formula><tex-math id=\"M5\">begin{document}$ n $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M6\">begin{document}$ t $end{document}</tex-math></inline-formula>. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect <inline-formula><tex-math id=\"M7\">begin{document}$ t $end{document}</tex-math></inline-formula>-error-correcting code in <inline-formula><tex-math id=\"M8\">begin{document}$ S_n $end{document}</tex-math></inline-formula> under the Hamming metric for some <inline-formula><tex-math id=\"M9\">begin{document}$ n $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M10\">begin{document}$ t = 1,2,3,4 $end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id=\"M11\">begin{document}$ 2t+1leq nleq max{4t^2e^{-2+1/t}-2,2t+1} $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M12\">begin{document}$ tgeq 2 $end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id=\"M13\">begin{document}$ min{frac{e}{2}sqrt{n+2},lfloorfrac{n-1}{2}rfloor}leq tleq lfloorfrac{n-1}{2}rfloor $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M14\">begin{document}$ ngeq 7 $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M15\">begin{document}$ e $end{document}</tex-math></inline-formula> is the Napier's constant.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89327565","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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