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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" -1 and h e a r span uL(s) where These linear recursive sequences suffer from one drawback: only relatively few terms of the sequence are needed to solve for the generating recursion; i.e., their linear span (the length of the shortest linear recursion over G F ( 2 ) satisfied by the sequence) is short relative to their period.Such easy predictability makes binary n-sequences unsuitable for some applications requiring pseudorandom bits.