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 begin{document}$ Z $end{document} and zero odd-period correlation zone (ZOCZ) width begin{document}$ Z_{odd} = Z $end{document} by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.
Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width begin{document}$ Z $end{document} and zero odd-period correlation zone (ZOCZ) width begin{document}$ Z_{odd} = Z $end{document} by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.
{"title":"Z-complementary pairs with flexible lengths and large zero odd-periodic correlation zones","authors":"Liqun Yao, Wenli Ren, Yong Wang, Chunming Tang","doi":"10.3934/amc.2021037","DOIUrl":"https://doi.org/10.3934/amc.2021037","url":null,"abstract":"<p style='text-indent:20px;'>Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width <inline-formula><tex-math id=\"M1\">begin{document}$ 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":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":"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}
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":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":"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}
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) $end{document} to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76577929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on begin{document}$ 2^m $end{document} variables are given by modifying the support of the quartic functions, where begin{document}$ m $end{document} is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.
In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on begin{document}$ 2^m $end{document} variables are given by modifying the support of the quartic functions, where begin{document}$ m $end{document} is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.
{"title":"A new construction of weightwise perfectly balanced Boolean functions","authors":"Rui Zhang, Sihong Su","doi":"10.3934/AMC.2021020","DOIUrl":"https://doi.org/10.3934/AMC.2021020","url":null,"abstract":"In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on begin{document}$ 2^m $end{document} variables are given by modifying the support of the quartic functions, where begin{document}$ m $end{document} is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74627807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ziteng Huang, Weijun Fang, Fang-Wei Fu, Fengting Li
Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square begin{document}$ q $end{document}, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of begin{document}$ q $end{document}-ary MDS Euclidean self-dual codes of lengths in the form begin{document}$ sfrac{q-1}{a}+tfrac{q-1}{b} $end{document}, where begin{document}$ s $end{document} and begin{document}$ t $end{document} range in some interval and begin{document}$ a, b ,|, (q -1) $end{document}. In particular, for large square begin{document}$ q $end{document}, our constructions take up a proportion of generally more than 34% in all the possible lengths of begin{document}$ q $end{document}-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.
Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square begin{document}$ q $end{document}, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of begin{document}$ q $end{document}-ary MDS Euclidean self-dual codes of lengths in the form begin{document}$ sfrac{q-1}{a}+tfrac{q-1}{b} $end{document}, where begin{document}$ s $end{document} and begin{document}$ t $end{document} range in some interval and begin{document}$ a, b ,|, (q -1) $end{document}. In particular, for large square begin{document}$ q $end{document}, our constructions take up a proportion of generally more than 34% in all the possible lengths of begin{document}$ q $end{document}-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.
{"title":"Generic constructions of MDS Euclidean self-dual codes via GRS codes","authors":"Ziteng Huang, Weijun Fang, Fang-Wei Fu, Fengting Li","doi":"10.3934/amc.2021059","DOIUrl":"https://doi.org/10.3934/amc.2021059","url":null,"abstract":"<p style='text-indent:20px;'>Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square <inline-formula><tex-math id=\"M1\">begin{document}$ 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 $end{document}</tex-math></inline-formula>-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76043707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let begin{document}$ mgeq3 $end{document} be a positive integer and begin{document}$ n = 2m $end{document}. Let begin{document}$ f(x) = x^{2^m+3} $end{document} be a power permutation over begin{document}$ {mathrm {GF}}(2^n) $end{document}, which is a monomial with a Niho exponent. In this paper, the differential spectrum of begin{document}$ f $end{document} is investigated. It is shown that the differential spectrum of begin{document}$ f $end{document} is begin{document}$ 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} $end{document} when begin{document}$ m $end{document} is even, and begin{document}$ 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} $end{document} when begin{document}$ m $end{document} is odd.
Let begin{document}$ mgeq3 $end{document} be a positive integer and begin{document}$ n = 2m $end{document}. Let begin{document}$ f(x) = x^{2^m+3} $end{document} be a power permutation over begin{document}$ {mathrm {GF}}(2^n) $end{document}, which is a monomial with a Niho exponent. In this paper, the differential spectrum of begin{document}$ f $end{document} is investigated. It is shown that the differential spectrum of begin{document}$ f $end{document} is begin{document}$ 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} $end{document} when begin{document}$ m $end{document} is even, and begin{document}$ 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} $end{document} when begin{document}$ m $end{document} is odd.
{"title":"Differential spectra of a class of power permutations with Niho exponents","authors":"Zhen Li, Haode Yan","doi":"10.3934/amc.2021060","DOIUrl":"https://doi.org/10.3934/amc.2021060","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ 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":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":"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":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":"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":"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":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":"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}
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} $. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.
在这个手稿中，我们研究了非链环$ mathcal{R} = frac{mathbb{F}_2[u]}{langle u^3 - urangle} $。让$ min mathbb{N} $和$ L, M, N subseteq [m]: = {1, 2, dots, m} $。对于$ Xsubseteq [m] $，定义$ Delta_X: = {v in mathbb{F}_2^m : text{Supp}(v)subseteq X} $和$ D: = (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3 $，它们是一个有序有限多集，由来自$ mathcal{R}^m $的元素组成，其中$ D_1in {Delta_L, Delta_L^c}, D_2in {Delta_M, Delta_M^c}, D_3in {Delta_N, Delta_N^c} $。对每个$ D $研究了$ {big(vcdot dbig)_{din D} : v in mathcal{R}^m } $定义的线性代码$ C_D $ over $ mathcal{R} $。此外，我们还考虑了具有两个极大元的简单复形。我们研究了它们的二值灰度图像和对应于$ mathcal{R} $的某个$ mathbb{F}_{2} $ -函数的二值类子域码。在每种情况下，得到了这些二元线性码最小且自正交的充分条件。此外，我们还得到了关于Griesmer界的无穷一族最优码。本文中得到的大多数代码都是小权重代码。
{"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} $. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135319496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a new class of generalized cyclotomic binary sequences with period begin{document}$ 4p^n $end{document} is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.
In this paper, a new class of generalized cyclotomic binary sequences with period begin{document}$ 4p^n $end{document} is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.
{"title":"On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $","authors":"Lin Yi, Xiangyong Zeng, Zhimin Sun, Shasha Zhang","doi":"10.3934/AMC.2021019","DOIUrl":"https://doi.org/10.3934/AMC.2021019","url":null,"abstract":"In this paper, a new class of generalized cyclotomic binary sequences with period begin{document}$ 4p^n $end{document} is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.","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":"88889992","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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