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A class of BCH codes with length $$frac{q^{2m}-1}{q+1}$$ 一类长度为 $$frac{q^{2m}-1}{q+1}$ 的 BCH 编码
IF 0.7 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-12-27 DOI: 10.1007/s00200-023-00637-z
{"title":"A class of BCH codes with length $$frac{q^{2m}-1}{q+1}$$","authors":"","doi":"10.1007/s00200-023-00637-z","DOIUrl":"https://doi.org/10.1007/s00200-023-00637-z","url":null,"abstract":"<h3>Abstract</h3> <p>As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length <span> <span>(frac{q^{2m}-1}{q+1})</span> </span> over the finite fields <span> <span>({mathbb {F}}_q)</span> </span>. We settle a conjecture about the largest <em>q</em>-cyclotomic coset leader modulo <em>n</em> which was proposed by Wu et al. We also get the second largest <em>q</em>-cyclotomic coset leader modulo <em>n</em> if <em>m</em> is odd. Moreover, we investigate the parameters of <span> <span>({mathcal {C}}_{(n,q,delta _i)})</span> </span> (<span> <span>({mathcal {C}}_{(n,q,delta _i)}^perp)</span> </span>) for <span> <span>(i=1,2)</span> </span> and <span> <span>({mathcal {C}}_{(n,q,delta )})</span> </span> (<span> <span>({mathcal {C}}_{(n,q,delta )}^perp)</span> </span>) for <span> <span>(2le delta le q-1)</span> </span>. What’s more, we obtain many (almost) optimal codes.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139062613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Construction of a class of at most three-weight linear codes and the applications 构建一类最多三权线性编码及其应用
IF 0.7 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-12-16 DOI: 10.1007/s00200-023-00638-y
Wenhui Liu, Xiaoni Du, Xingbin Qiao
{"title":"Construction of a class of at most three-weight linear codes and the applications","authors":"Wenhui Liu, Xiaoni Du, Xingbin Qiao","doi":"10.1007/s00200-023-00638-y","DOIUrl":"https://doi.org/10.1007/s00200-023-00638-y","url":null,"abstract":"<p>Linear codes are widely studied due to their important applications in authentication codes, association schemes and strongly regular graphs, etc. In this paper, a class of at most three-weight linear codes is constructed by selecting a new defining set, then the parameters and weight distributions of codes are determined by exponential sums. Results show that almost all the linear codes we constructed are minimal and we also describe the access structures of the secret sharing schemes based on their dual. Especially, the new binary code is a two-weight projective code and based on which a strongly regular graph with new parameters is designed.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138679772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Several classes of optimal cyclic codes with three zeros 有三个零的几类最优循环码
IF 0.7 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-12-07 DOI: 10.1007/s00200-023-00636-0
Tingting Wu, Li Liu, Lanqiang Li
{"title":"Several classes of optimal cyclic codes with three zeros","authors":"Tingting Wu, Li Liu, Lanqiang Li","doi":"10.1007/s00200-023-00636-0","DOIUrl":"https://doi.org/10.1007/s00200-023-00636-0","url":null,"abstract":"<p>As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal <i>p</i>-ary cyclic codes with parameters <span>([p^m-1, p^m-frac{3m}{2}-2, 4])</span> by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code <span>(mathcal {C}_{(0,1,e)})</span> and present three classes of optimal codes with parameters <span>([7^m-1,7^m-2m-2,4])</span>. Additionally, we provide the weight distribution of one class of their duals.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138563195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A special scalar multiplication algorithm on Jacobi quartic curves Jacobi四次曲线上的一种特殊标量乘法算法
IF 0.7 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-11-16 DOI: 10.1007/s00200-023-00633-3
Jiang Weng, Aiwang Chen, Tao Huang, Weifeng Ji
{"title":"A special scalar multiplication algorithm on Jacobi quartic curves","authors":"Jiang Weng, Aiwang Chen, Tao Huang, Weifeng Ji","doi":"10.1007/s00200-023-00633-3","DOIUrl":"https://doi.org/10.1007/s00200-023-00633-3","url":null,"abstract":"<p>At present, GLV/GLS scalar multiplication mainly focuses on elliptic curves in Weierstrass form, attempting to find and construct more and more efficiently computable endomorphism. In this paper, we investigate the application of the GLV/GLS scalar multiplication technique to Jacobi Quartic curves. Firstly, we present a concrete construction of efficiently computable endomorphisms for this type of curves over prime fields by exploiting birational equivalence between curves, and obtain a 2-dimensional GLV method. Secondly, we consider the quadratic twists of elliptic curves. By using birational equivalence and Frobenius mapping between curves, we present methods to construct efficiently computable endomorphisms for this type of curves over the quadratic extension field, and obtain a 2-dimensional GLS method. Finally, we obtain a 4-dimensional GLV method on elliptic curves with <i>j</i>-invariant 0 or 1728 by using higher degree twists. The experimental results show that the speedups of the 2-dimensional GLV method and 4-dimensional GLV method compared to 5-NAF method exceed <span>(37.2%)</span> and <span>(109.4%)</span> for Jacobi Quartic curves, respectively. At the same time, when utilizing one of the proposed methods, the scalar multiplication on Jacobi Quartic curves is consistently more efficient than on elliptic curves in Weierstrass form.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Using elliptic curves to construct 3D arrays 利用椭圆曲线构造三维阵列
4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-11-02 DOI: 10.1007/s00200-023-00634-2
Alcibíades Bustillo-Zárate, Dorothy Bollman, José Ortiz-Ubarri
{"title":"Using elliptic curves to construct 3D arrays","authors":"Alcibíades Bustillo-Zárate, Dorothy Bollman, José Ortiz-Ubarri","doi":"10.1007/s00200-023-00634-2","DOIUrl":"https://doi.org/10.1007/s00200-023-00634-2","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135934743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalized characteristic sets and new multivariate difference dimension polynomials 广义特征集和新的多元差维多项式
IF 0.6 4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-10-28 DOI: 10.1007/s00200-023-00628-0
Alexander Levin
{"title":"Generalized characteristic sets and new multivariate difference dimension polynomials","authors":"Alexander Levin","doi":"10.1007/s00200-023-00628-0","DOIUrl":"10.1007/s00200-023-00628-0","url":null,"abstract":"<div><p>We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials and a partition of the basic set of translations <span>(sigma)</span>. Using properties of these characteristic sets, we prove the existence and outline a method of computation of a multivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of <span>(sigma)</span> are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than previously known difference dimension polynomials. In particular, a dimension polynomial of the new type associated with a system of algebraic difference equations gives more information about the system than the classical univariate difference dimension polynomial.</p></div>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Frames over finite fields and self-dual codes 有限域上的帧和自对偶码
4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-10-25 DOI: 10.1007/s00200-023-00630-6
Minjia Shi, Yingying Liu, Jon-Lark Kim, Patrick Solé
{"title":"Frames over finite fields and self-dual codes","authors":"Minjia Shi, Yingying Liu, Jon-Lark Kim, Patrick Solé","doi":"10.1007/s00200-023-00630-6","DOIUrl":"https://doi.org/10.1007/s00200-023-00630-6","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135217866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Further results on the $$(-1)$$-differential uniformity of some functions over finite fields with odd characteristic 若干函数在奇特征有限域上$$(-1)$$ -微分均匀性的进一步结果
4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-10-21 DOI: 10.1007/s00200-023-00632-4
Qian Liu, Ximeng Liu, Meixiang Chen, Jian Zou, Zhiwei Huang
{"title":"Further results on the $$(-1)$$-differential uniformity of some functions over finite fields with odd characteristic","authors":"Qian Liu, Ximeng Liu, Meixiang Chen, Jian Zou, Zhiwei Huang","doi":"10.1007/s00200-023-00632-4","DOIUrl":"https://doi.org/10.1007/s00200-023-00632-4","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
MDS codes based on orthogonality of quasigroups 基于拟群正交性的MDS码
4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-10-20 DOI: 10.1007/s00200-023-00631-5
Satish Kumar, Harshdeep Singh, Indivar Gupta, Ashok Ji Gupta
{"title":"MDS codes based on orthogonality of quasigroups","authors":"Satish Kumar, Harshdeep Singh, Indivar Gupta, Ashok Ji Gupta","doi":"10.1007/s00200-023-00631-5","DOIUrl":"https://doi.org/10.1007/s00200-023-00631-5","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Nonlinear complexity from the Hermitian and the Suzuki function fields 厄米和铃木函数域的非线性复杂性
4区 工程技术
Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-10-04 DOI: 10.1007/s00200-023-00629-z
Ferruh Özbudak, Nesrin Tutaş
{"title":"Nonlinear complexity from the Hermitian and the Suzuki function fields","authors":"Ferruh Özbudak, Nesrin Tutaş","doi":"10.1007/s00200-023-00629-z","DOIUrl":"https://doi.org/10.1007/s00200-023-00629-z","url":null,"abstract":"","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135551898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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