On the linear span of binary sequences obtained from finite Geometries

Abstract

A class of periodic binary sequences that are obtained from the incidence vectors of hyperplanes in finite geometries is defined, and a general method to determine their linear spans (the length of the shortest linear recursion over GF(2) satisfied by the sequence) is described. In particular, we show that the projective and affine hyperplane sequences of odd order both have full linear span. Another application involves the parity sequence of order n, which has period p n − 1 and linear span vL(s) where v = (p n − 1)/(p − 1) and L(s) is the linear span of a parity sequence of order 1. The determination of the linear span of the parity sequence of order 1 leads to an interesting open problem involving primes.