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

Content Provider	Springer Nature Link
Author	Fu, Fang Wei Li, Chengju Yue, Qin
Copyright Year	2016
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 $${\mathcal {C}}_D$$ .
Ending Page	30
Page Count	20
Starting Page	11
File Format	PDF
ISSN	09381279
e-ISSN	14320622
Journal	Applicable Algebra in Engineering, Communication and Computing
Issue Number	1
Volume Number	28
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-08-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Gauss sums Computer Hardware Linear codes Weight distributions Artificial Intelligence (incl. Robotics) Linear codes, general Algebraic coding theory; cryptography Theory of Computation Exponential sums Symbolic and Algebraic Manipulation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Algebra and Number Theory

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