On the Weight Distributions of Some \(q\)-ary Cyclic Codes

Abstract

Let \(p\) be an odd prime, \(q\) be a prime power coprime to \(p\), and \(n\) be a positive integer. For any positive integer \(d \leq n\), let \(g_1(x) = {x^{p^{n-d}} – 1}\),\(g_2(x)=1+{x^{p^{n – d+1}}}+x^{2p^{n-d+1}}+ \ldots +x^{(p^{d-1}-1)p^{n-d+1}}\),and , \(g_3(x) =1+x^{p^{n-d}}+x^{2p^{n-d}}+ \ldots +x^{(p-1)p^{n-d}} \). In this paper, we determine the weight distributions of \(q\)-ary cyclic codes of length \(pn\) generated by the polynomials \(g_1(x)\), \(g_2(x)\), \(g_3(x)\), \(g_4(x)\), and \(g_5(x)\), by employing the techniques developed in Sharma \& Bakshi [11]. Keywords: cyclic codes, Hamming weight, weight spectrum.