Parameters of hulls of primitive binary narrow-sense BCH codes and their subcodes

Abstract

<p style='text-indent:20px;'>The (Euclidean) hull of a linear code is defined to be the intersection of the code and its Euclidean dual. It is clear that the hulls are self-orthogonal codes, which are an important type of linear codes due to their wide applications in communication and cryptography. Let <inline-formula><tex-math id="M1">\begin{document}$ \mathcal C_{(2,n,\delta)} $\end{document}</tex-math></inline-formula> be the binary primitive narrow-sense BCH code, where <inline-formula><tex-math id="M2">\begin{document}$ n = 2^m-1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> is a positive integer. In this paper, we will investigate the parameters of the hulls of <inline-formula><tex-math id="M4">\begin{document}$ \mathcal C_{(2,n,\delta)} $\end{document}</tex-math></inline-formula>. The dimension of <inline-formula><tex-math id="M5">\begin{document}$ \text{Hull}(\mathcal C_{(2,n,\delta)}) $\end{document}</tex-math></inline-formula> will be presented when <inline-formula><tex-math id="M6">\begin{document}$ 2 \le \delta \le 2^{\lfloor \frac{m}{2} \rfloor+1}+2^{\lceil \frac{m}{2} \rceil-1}-1 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ m \ge 5 $\end{document}</tex-math></inline-formula>. Furthermore, we give an improvement on lower bounds of the minimum distances of <inline-formula><tex-math id="M8">\begin{document}$ \text{Hull}(\mathcal C_{(2,n,\delta)}) $\end{document}</tex-math></inline-formula>. We also construct a self-orthogonal subcode of <inline-formula><tex-math id="M9">\begin{document}$ \text{Hull}(\mathcal C_{(2,n,\delta)}) $\end{document}</tex-math></inline-formula> and investigate the parameters of the self-orthogonal code.</p>