The polynomial degree of the Grassmannian G(1,n,q) of lines in finite projective space PG(n, q)

Abstract

Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with $N:=\binom{n+1}{2}-1.$ For n = 2 or 3 the characteristic function $\chi (\overline{G})$ of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of $\chi (\overline{G})$ is $(q-1)(\binom{n}{2}-1+\delta )$ for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg($\chi (\overline{G}))= \binom{n}{2}-1$ for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least $\binom{n}{2}.$