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 \(d \leq n\), let \(g_1(x) = 1 + x^{p^{n-d}} + x^{{2p}^{n-d}} + \ldots + x^{(p-1)p^{n-d}}\) and \(g_2(x) = 1 + x^{p^{n-d+1}} + x^{2p^{n-d+1}} + \ldots + x^{{(p^{d-1}-1)}{p^{n-d+1}}}\). 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 \(g_01(x)g_2(x)\), which is effective for small values of \(p\) and \(d\).