A new class of distance-optimal binary cyclic codes and their duals

Abstract

Let $$m=8k$$ and $$\alpha$$ be a primitive element of the finite field $${{\mathbb {G}}{\mathbb {F}}}(2^m)$$ , where $$k\ge 2$$ is an integer. In this paper, a class of binary cyclic codes $${{\mathcal {C}}}_{(u,v)}$$ of length $$2^m-1$$ with two nonzeros $$\alpha ^{-u}$$ and $$\alpha ^{-v}$$ is studied, where $$(u,v)=(1,(2^{m}-1)/17)$$ . It turns out that $${{\mathcal {C}}}_{(u,v)}$$ has parameters $$[2^m-1,2^m-m-9,4]$$ and is distance-optimal with respect to the Sphere Packing bound. The weight distribution of the dual of $${{\mathcal {C}}}_{(u,v)}$$ is also completely determined based on some results on Gaussian periods.