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

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Abstract

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 \({\mathcal {C}}_D\).

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Acknowledgments

The authors are very grateful to the editor and the anonymous reviewers for their valuable comments and suggestions that improved the quality of this paper. The paper is supported by the National Natural Science Foundation of China (Nos. 11171150, 61571243, and 61171082), the Fundamental Research Funds for the Central Universities (No. 56XZA15002), and the 973 Program of China (Grant No. 2013CB834204).

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Authors and Affiliations

  1. School of Computer Science and Software Engineering, East China Normal University, 200062, Shanghai, China

    Chengju Li

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, 211100, Nanjing, People’s Republic of China

    Qin Yue

  3. Chern Institute of Mathematics and LPMC, Nankai University, 300071, Tianjin, China

    Fang-Wei Fu

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  1. Chengju Li
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  2. Qin Yue
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  3. Fang-Wei Fu
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Correspondence to Chengju Li.

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Li, C., Yue, Q. & Fu, FW. A construction of several classes of two-weight and three-weight linear codes. AAECC 28, 11–30 (2017). https://doi.org/10.1007/s00200-016-0297-4

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  • DOI: https://doi.org/10.1007/s00200-016-0297-4

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