A construction of several classes of two-weight and three-weight linear codes

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• 作者：Chengju Li ; Qin Yue ; Fang-Wei Fu
• 关键词：Linear codes ; Weight distributions ; Gauss sums
• 刊名：Applicable Algebra in Engineering, Communication and Computing
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：28
• 期：1
• 页码：11-30
• 全文大小：
• 刊物类别：Computer Science
• 刊物主题：Symbolic and Algebraic Manipulation; Computer Hardware; Theory of Computation; Artificial Intelligence (incl. Robotics);
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0622
• 卷排序：28

文摘

Linear codes constructed from defining sets have been extensively studied and may have a few nonzero weights if the defining sets are well chosen. Let \({\mathbb {F}}_q\) be a finite field with \(q=p^m\) elements, where p is a prime and m is a positive integer. Motivated by Ding and Ding’s recent work (IEEE Trans Inf Theory 61(11):5835–5842, 2015), we construct p-ary linear codes \({\mathcal {C}}_D\) by $$\begin{aligned} {\mathcal {C}}_D=\{{\mathbf {c}}(a,b)=\big (\text {Tr}_m(ax+by)\big )_{(x,y)\in D}: a, b \in {\mathbb {F}}_q\}, \end{aligned}$$where \(D \subset {\mathbb {F}}_q^2\) and \(\text {Tr}_m\) is the trace function from \({\mathbb {F}}_q\) onto \({\mathbb {F}}_p\). In this paper, we will employ exponential sums to investigate the weight enumerators of the linear codes \({\mathcal {C}}_D\), where \(D=\{(x, y) \in {\mathbb {F}}_q^2 \setminus \{(0,0)\}: \text {Tr}_m(x^{N_1}+y^{N_2})=0\}\) for two positive integers \(N_1\) and \(N_2\). Several classes of two-weight and three-weight linear codes and their explicit weight enumerators are presented if \(N_1, N_2 \in \{1, 2, p^{\frac{m}{2}}+1\}\). By deleting some coordinates, more punctured two-weight and three-weight linear codes \({\mathcal {C}}_{\overline{D}}\) which include some optimal codes are derived from \({\mathcal {C}}_D\).KeywordsLinear codesWeight distributionsGauss sumsMathematics Subject Classification94B0511T7111T23References1.Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums. Wiley, New York (1997)MATHGoogle Scholar2.Calderbank, A.R., Goethals, J.M.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984)MATHMathSciNetGoogle Scholar3.Calderbank, A.R., Kantor, W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986)CrossRefMATHMathSciNetGoogle Scholar4.Carlet, C., Ding, C., Yuan, J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. 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(2015). doi:10.1007/s10623-015-0144-9 Copyright information© Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsChengju Li1Email authorQin Yue2Fang-Wei Fu31.School of Computer Science and Software EngineeringEast China Normal UniversityShanghaiChina2.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China3.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina About this article CrossMark Publisher Name Springer Berlin Heidelberg Print ISSN 0938-1279 Online ISSN 1432-0622 About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips