Constructions for several classes of few-weight linear codes and their applications

Abstract

<p style='text-indent:20px;'>In this paper, for any odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and an integer <inline-formula><tex-math id="M2">\begin{document}$ m\ge 3 $\end{document}</tex-math></inline-formula>, several classes of linear codes with <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-weight <inline-formula><tex-math id="M4">\begin{document}$ (t = 3,5,7) $\end{document}</tex-math></inline-formula> are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for <inline-formula><tex-math id="M5">\begin{document}$ m = 3 $\end{document}</tex-math></inline-formula>, a class of MDS codes with parameters <inline-formula><tex-math id="M6">\begin{document}$ [p,3,p-2] $\end{document}</tex-math></inline-formula> are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and <inline-formula><tex-math id="M7">\begin{document}$ s $\end{document}</tex-math></inline-formula>-sum sets for any odd <inline-formula><tex-math id="M8">\begin{document}$ s>1 $\end{document}</tex-math></inline-formula>.</p>