Correlation Functions of Geometric Sequences

Abstract

This paper considers the cross-correlation function values of a family of binary sequences obtained born finite geometries.These values are shown to depend on the intersection of hyperplanes in a projective space and the cross-correlation function values of the nonlinear feedforward functions used in the construction of the geometric sequences.

Cross-correlations of geometric sequences in characteristic two

“…Notable early examples are GMW sequences [6] and bent sequences [14]. More general sequences, in which a nonlinear feedforward function is applied to an m-sequence over a finite field, have been studied in the past decade by a number of authors, for example [1,2,3,4,7,8]. We call these sequences geometric sequences.…”

“…However, their cross-correlations are known in only a small number of cases, and their linear complexities are far from the maximum possible for arbitrary sequences. The author (with Chan and Goresky) has previously considered cross-correlation function values of pairs of geometric sequences that are obtained from the same q-ary m-sequence but different nonlinear feedforward functions [4] and of geometric sequences in characteristic two whose underlying m-sequences differ by a quadratic decimation [7]. (A quadratic decimation is a k-fold decimation -every k-th element -where the sum of the coefficients of the base q expansion of k equals two.…”

Ternary perfect sequences with three-valued cross-correlation

“…They are also closely related to some mathematical objects such as Hadamard matrices, difference sets, and bent functions [8]. During these decades, a vast of algebraic or combinatorial constructions of sequences with low correlation have been reported in the literature (see [1], [4], [5], [6], [8], [10], [13], [12], [14], [22], [23], [24], and the references therein).…”