Two classes of binary cyclic codes and their weight distributions (Q2032300)

The authors of this study work with the following cyclic codes \(C_{(u,v)}=\{(\mathrm{Tr}_1^{l_u}(ax^u)+\mathrm{Tr}_1^{l_v}(bx^v))_{x\in\mathbb{F}_q^\ast}\mid a\in\mathbb{F}_{2^{l_u}}, b\in\mathbb{F}_{2^{l_v}}\}\) where \(\mathrm{Tr}_1^{l_u}\) denotes the trace function from \(\mathbb{F}_{2^{l_u}}\) to \(\mathbb{F}_2\). Two classes of cyclic codes with a few weights are presented and the weight distributions of these cyclic codes are settled for two cases: 1) \(m=4k\) and \((u,v)=(1,\frac{2^m-1}{5});\) 2) \(m=6k\) and \((u,v)=(1,\frac{2^m-1}{9}).\) The duals of the first class of cyclic codes are also studied. In the case of the cyclic code \(C_{(u,v)}\) for \((u,v)=(1,\frac{2^m-1}{5})\), its duals are established as \([2^m-1,2^m-m-5,d]\) codes where \(4\leq d\leq 6.\)