Jacobi Sums and Correlations of Sidelnikov Sequences

Abstract

We consider the problem of determining the cross-correlation values of the sequences in the families comprised of constant multiples of $M$-ary Sidelnikov sequences over ${\mathbb{F}}_q$, where $q$ is a power of an odd prime $p$. We show that the cross-correlation values of pairs of sequences from such a family can be expressed in terms of certain Jacobi sums. This insight facilitates the computation of the cross-correlation values of these sequence pairs so long as $\varphi (M{)}^{\varphi (M)}\le q.$ We are also able to use our Jacobi sum expression to deduce explicit formulae for the cross-correlation distribution of a family of this type in the special case that there exists an integer $x$ such that $p^x\equiv -1(\mathrm{mod}M).$

arXiv admin note: text overlap with arXiv:1809.04010