Results on the Crosscorrelation and Autocorrelation of Sequences

Abstract

In this paper we investigate some properties of the crosscorrelation spectrum of an m-sequence a with period 2^m − 1 and a d-decimation b. Recently, Lahtonen et. al. [1] calculated the crosscorrelation value Θ _d (1) for specific exponents (i.e. for Gold and Kasami type). In this paper we generalize this result to all known almost bent exponents. In [1], the authors also prove that Gold functions are bent on some hyperplanes with respect to the base field $\mathbb{F}{2}$. We also generalize this result and show that Gold functions are bent on all hyperplanes with respect to any subfield $\mathbb{F}{2^k}$. We also show that $\Theta_d(1) \ne 0$ for many exponents d, and conclude that many sequences of type a + b (including m-sequences added to an almost bent decimation) do not have perfect autocorrelation.