The Weight Distributions of Some Binary Cyclic Codes

Abstract

Let $p$ be an odd prime and $n$ be a positive integer. For any positive integer $d\le n$, let $g_1(x)=1+x^{p^{n-d}}+x^{{2p}^{n-d}}+\dots +x^{(p-1)p^{n-d}}$ and $g_2(x)=1+x^{p^{n-d+1}}+x^{2p^{n-d+1}}+\dots +x^{(p^{d-1}-1)p^{n-d+1}}$. In this paper, we provide a method to determine the weight distributions of binary cyclic codes of length $p^n$ generated by the polynomials $g_1(x)$ and $g_{0}1(x)g_2(x)$, which is effective for small values of $p$ and $d$.