The geometry of m-Sequences: Three-valued crosscorrelations and quadrics in finite projective geometry

Hyperplanes H and sets $H^r $ of $PG ( n - 1,2 )$ are identified with pairs of binary m-sequences of span n. If $H^r $ is a quadric, then a three-valued periodic crosscorrelation function between the m-sequences results. Conjectures concerning three-valued periodic crosscorrelation functions of binary m-sequences specialize to conjectures concerning the degeneracy of quadrics of the form $H^r $. The main result is that if $n = 2^k m$, with m odd and $k\geqq 2$, $H \subseteq PG ( n - 1,2 )$ is a hyperplane and $H^r $ is a quadric, necessarily a cone of order $2l + 1$, then $2l + 1\geqq 2^{k - 1} + 1$. This shows that when $n \equiv 0 ( \bmod 4 )$, there are no m-sequences arising from quadrics with preferred three-valued periodic crosscorrelation functions. Also, when $n = 2^k $, m-sequences arising from quadrics would have three-valued periodic crosscorrelation functions with values determined by a cone of order at least $( n / 2 ) + 1$.