On binary LCD BCH codes of length $ \frac{{{2^m} + 1}}{3} $

Abstract

<p style='text-indent:20px;'>BCH codes are a special subclass of cyclic codes and have many important applications in data storage and communication systems. In this paper, we investigate the structure of binary linear complementary dual (LCD) BCH codes with length <inline-formula><tex-math id="M2">\begin{document}$ n = \frac{{{2^m} + 1}}{3} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ m \geq 7 $\end{document}</tex-math></inline-formula> is an odd integer. By exploring cyclotomic cosets modulo <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>, we determine the dimension of LCD BCH codes for designed distance <inline-formula><tex-math id="M5">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> in the range <inline-formula><tex-math id="M6">\begin{document}$ 2 \le \delta \le{2^{\frac{{m + 1}}{2}}} $\end{document}</tex-math></inline-formula>. Furthermore, we compute the first five largest coset leaders modulo <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula> and construct some binary LCD BCH codes. We also present two families of optimal binary linear codes from LCD BCH codes.</p>