Advances in Mathematics of Communications最新文献

{"title":"Generic constructions of MDS Euclidean self-dual codes via GRS codes","authors":"Ziteng Huang, Weijun Fang, Fang-Wei Fu, Fengting Li","doi":"10.3934/amc.2021059","DOIUrl":"https://doi.org/10.3934/amc.2021059","url":null,"abstract":"<p style='text-indent:20px;'>Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square <inline-formula><tex-math id=\"M1\">begin{document}$ q $end{document}</tex-math></inline-formula>, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of <inline-formula><tex-math id=\"M2\">begin{document}$ q $end{document}</tex-math></inline-formula>-ary MDS Euclidean self-dual codes of lengths in the form <inline-formula><tex-math id=\"M3\">begin{document}$ sfrac{q-1}{a}+tfrac{q-1}{b} $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M4\">begin{document}$ s $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M5\">begin{document}$ t $end{document}</tex-math></inline-formula> range in some interval and <inline-formula><tex-math id=\"M6\">begin{document}$ a, b ,|, (q -1) $end{document}</tex-math></inline-formula>. In particular, for large square <inline-formula><tex-math id=\"M7\">begin{document}$ q $end{document}</tex-math></inline-formula>, our constructions take up a proportion of generally more than 34% in all the possible lengths of <inline-formula><tex-math id=\"M8\">begin{document}$ q $end{document}</tex-math></inline-formula>-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"106 1","pages":"1453-1467"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76043707","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":"Differential spectra of a class of power permutations with Niho exponents","authors":"Zhen Li, Haode Yan","doi":"10.3934/amc.2021060","DOIUrl":"https://doi.org/10.3934/amc.2021060","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ mgeq3 $end{document}</tex-math></inline-formula> be a positive integer and <inline-formula><tex-math id=\"M2\">begin{document}$ n = 2m $end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id=\"M3\">begin{document}$ f(x) = x^{2^m+3} $end{document}</tex-math></inline-formula> be a power permutation over <inline-formula><tex-math id=\"M4\">begin{document}$ {mathrm {GF}}(2^n) $end{document}</tex-math></inline-formula>, which is a monomial with a Niho exponent. In this paper, the differential spectrum of <inline-formula><tex-math id=\"M5\">begin{document}$ f $end{document}</tex-math></inline-formula> is investigated. It is shown that the differential spectrum of <inline-formula><tex-math id=\"M6\">begin{document}$ f $end{document}</tex-math></inline-formula> is <inline-formula><tex-math id=\"M7\">begin{document}$ mathbb S = {omega_0 = 2^{2m-1}+2^{2m-3}-1,omega_2 = 2^{2m-2}+2^{m-1}, omega_4 = 2^{2m-3}-2^{m-1},omega_{2^m} = 1} $end{document}</tex-math></inline-formula> when <inline-formula><tex-math id=\"M8\">begin{document}$ m $end{document}</tex-math></inline-formula> is even, and <inline-formula><tex-math id=\"M9\">begin{document}$ mathbb S = {omega_0 = frac{7cdot2^{2m-2}+2^m}3, omega_2 = 3cdot2^{2m-3}-2^{m-2}-1, omega_6 = frac{2^{2m-3}-2^{m-2}}3, omega_{2^m+2} = 1} $end{document}</tex-math></inline-formula> when <inline-formula><tex-math id=\"M10\">begin{document}$ m $end{document}</tex-math></inline-formula> is odd.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"115 1","pages":"1468-1475"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76089295","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 over the cross product of the cyclic groups of order 2","authors":"S. Dougherty, S. Șahinkaya","doi":"10.3934/amc.2023005","DOIUrl":"https://doi.org/10.3934/amc.2023005","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"30 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74027386","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":"A new construction of weightwise perfectly balanced Boolean functions","authors":"Rui Zhang, Sihong Su","doi":"10.3934/AMC.2021020","DOIUrl":"https://doi.org/10.3934/AMC.2021020","url":null,"abstract":"In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on begin{document}$ 2^m $end{document} variables are given by modifying the support of the quartic functions, where begin{document}$ m $end{document} is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"61 1","pages":"757-770"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74627807","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":"Certain binary minimal codes constructed using simplicial complexes","authors":"Vidya Sagar, Ritumoni Sarma","doi":"10.3934/amc.2023044","DOIUrl":"https://doi.org/10.3934/amc.2023044","url":null,"abstract":"In this manuscript, we work over the non-chain ring $ mathcal{R} = frac{mathbb{F}_2[u]}{langle u^3 - urangle} $. Let $ min mathbb{N} $ and let $ L, M, N subseteq [m]: = {1, 2, dots, m} $. For $ Xsubseteq [m] $, define $ Delta_X: = {v in mathbb{F}_2^m : text{Supp}(v)subseteq X} $ and $ D: = (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3 $, an ordered finite multiset consisting of elements from $ mathcal{R}^m $, where $ D_1in {Delta_L, Delta_L^c}, D_2in {Delta_M, Delta_M^c}, D_3in {Delta_N, Delta_N^c} $. The linear code $ C_D $ over $ mathcal{R} $ defined by $ {big(vcdot dbig)_{din D} : v in mathcal{R}^m } $ is studied for each $ D $. Further, we also consider simplicial complexes with two maximal elements. We study their binary Gray images and the binary subfield-like codes corresponding to a certain $ mathbb{F}_{2} $-functional of $ mathcal{R} $. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"122 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":"135319496","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":"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":"20 1","pages":"1440-1452"},"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}

{"title":"Boolean function classes with high nonlinearity","authors":"Kezia Saini, M. Garg","doi":"10.3934/amc.2023014","DOIUrl":"https://doi.org/10.3934/amc.2023014","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81268903","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":"Z-complementary pairs with flexible lengths and large zero odd-periodic correlation zones","authors":"Liqun Yao, Wenli Ren, Yong Wang, Chunming Tang","doi":"10.3934/amc.2021037","DOIUrl":"https://doi.org/10.3934/amc.2021037","url":null,"abstract":"<p style='text-indent:20px;'>Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width <inline-formula><tex-math id=\"M1\">begin{document}$ Z $end{document}</tex-math></inline-formula> and zero odd-period correlation zone (ZOCZ) width <inline-formula><tex-math id=\"M2\">begin{document}$ Z_{odd} = Z $end{document}</tex-math></inline-formula> by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"23 1","pages":"1060-1071"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86895935","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 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":"26 1","pages":"920-927"},"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}

{"title":"Several classes of new projective three-weight or four-weight linear codes and their applications in $ s $-sum sets","authors":"","doi":"10.3934/amc.2023013","DOIUrl":"https://doi.org/10.3934/amc.2023013","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"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":"75814955","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}