The polynomial degree of the Grassmannian G(1,n,q) of lines in finite projective space PG(n, q)期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

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

Authors:	David G Glynn Johannes G Maks Rey Casse

Institution:	(1) School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia;(2) Department of Mathematics, Delft University of Technology, Delft, The Netherlands

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 $\binom{n}{2}.$

Keywords:	Polynomial degree Grassmannian G(1 n q) Geometric codes Quantum codes
本文献已被 SpringerLink 等数据库收录！