Five-weight codes from three-valued correlation of M-sequences

Abstract

<p style='text-indent:20px;'>In this paper, for each of six families of three-valued <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring <inline-formula><tex-math id="M2">\begin{document}$ R = \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ u^2 = 0. $\end{document}</tex-math></inline-formula> The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are <inline-formula><tex-math id="M4">\begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $\end{document}</tex-math></inline-formula> for some explicit <inline-formula><tex-math id="M5">\begin{document}$ r. $\end{document}</tex-math></inline-formula></p>