On the correlation distribution of Delsarte–Goethals sequences

Abstract

For odd integer m ≥ 3 and ${t=0,1,\ldots,\frac{m-1}{2}}$, we define Family ${\fancyscript{V}(t)}$ to be a set of size 2^m(t+1) containing binary sequences of period 2^m+1 − 2. The nontrivial correlations between sequences in Family ${\fancyscript{V}(t)}$ are bounded in magnitude by 2 + 2^(m+1)/2+t. Families ${\fancyscript{V}(0)}$ and ${\fancyscript{V}(1)}$ compare favourably to the small and large Kasami sets, respectively. So far, the correlation distribution of Family ${\fancyscript{V}(t)}$ is only known for t = 0. A general framework for computing the correlation distribution of Family ${\fancyscript{V}(t)}$ is established. The correlation distribution of ${\fancyscript{V}(1)}$ is derived, and a way to obtain the correlation distribution of ${\fancyscript{V}(2)}$ is described.