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 $-divisible quasi-cyclic codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the begin{document}$ 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":"33 1","pages":"572-588"},"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}
{"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":"44 1","pages":""},"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}
{"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":"41 1","pages":""},"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}
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":"34 1","pages":"815-827"},"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":"77 1","pages":""},"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}
Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco
{"title":"New bounds for covering codes of radius 3 and codimension $ 3t+1 $","authors":"Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco","doi":"10.3934/amc.2023042","DOIUrl":"https://doi.org/10.3934/amc.2023042","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A generalized Stirling numbers of the second kind begin{document}$ S_{a,b}left(p,kright) $end{document} , involved in the expansion begin{document}$ left(an+bright)^{p} = sum_{k = 0}^{p}k!S_{a,b}left(p,kright) binom{n}{k} $end{document} , where begin{document}$ a neq 0, b $end{document} are complex numbers, have studied in [ 16 ]. In this paper, we show that Bernoulli polynomials begin{document}$ B_{p}(x) $end{document} can be written in terms of the numbers begin{document}$ S_{1,x}left(p,kright) $end{document} , and then use the known results for begin{document}$ S_{1,x}left(p,kright) $end{document} to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.
A generalized Stirling numbers of the second kind begin{document}$ S_{a,b}left(p,kright) $end{document} , involved in the expansion begin{document}$ left(an+bright)^{p} = sum_{k = 0}^{p}k!S_{a,b}left(p,kright) binom{n}{k} $end{document} , where begin{document}$ a neq 0, b $end{document} are complex numbers, have studied in [ 16 ]. In this paper, we show that Bernoulli polynomials begin{document}$ B_{p}(x) $end{document} can be written in terms of the numbers begin{document}$ S_{1,x}left(p,kright) $end{document} , and then use the known results for begin{document}$ S_{1,x}left(p,kright) $end{document} to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.
{"title":"Several formulas for Bernoulli numbers and polynomials","authors":"T. Komatsu, Bijan Kumar Patel, C. Pita-Ruiz","doi":"10.3934/AMC.2021006","DOIUrl":"https://doi.org/10.3934/AMC.2021006","url":null,"abstract":"A generalized Stirling numbers of the second kind begin{document}$ S_{a,b}left(p,kright) $end{document} , involved in the expansion begin{document}$ left(an+bright)^{p} = sum_{k = 0}^{p}k!S_{a,b}left(p,kright) binom{n}{k} $end{document} , where begin{document}$ a neq 0, b $end{document} are complex numbers, have studied in [ 16 ]. In this paper, we show that Bernoulli polynomials begin{document}$ B_{p}(x) $end{document} can be written in terms of the numbers begin{document}$ S_{1,x}left(p,kright) $end{document} , and then use the known results for begin{document}$ S_{1,x}left(p,kright) $end{document} to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"40 1","pages":"522-535"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76577929","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}
Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in code division multiple access communication systems. In this paper, several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound. Specially, a class of optimal codebooks with respect to the Levenshtein bound is obtained. These classes of codebooks are constructed by selecting certain rows deterministically from circulant matrices, Fourier matrices and Hadamard matrices, respectively. The construction methods and parameters of some codebooks provided in this paper are new.
{"title":"Constructions of asymptotically optimal codebooks with respect to Welch bound and Levenshtein bound","authors":"Gang Wang, Deng-Ming Xu, Fang-Wei Fu","doi":"10.3934/amc.2021065","DOIUrl":"https://doi.org/10.3934/amc.2021065","url":null,"abstract":"Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in code division multiple access communication systems. In this paper, several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound. Specially, a class of optimal codebooks with respect to the Levenshtein bound is obtained. These classes of codebooks are constructed by selecting certain rows deterministically from circulant matrices, Fourier matrices and Hadamard matrices, respectively. The construction methods and parameters of some codebooks provided in this paper are new.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"114 1","pages":"1526-1541"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77721800","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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