The quintic Grassmannian g(1,4,2) in PG(9,2)

Shaw, R; Gordon, Neil

Abstract

The 155 points of the Grassmannian g(1,4,2) of lines of PG(4,2) = PV(5,2) are those points x is an element of PG(9,2) = P(boolean AND V-2(5,2)) which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X subset of PG(9,2) will be termed odd or even according as X intersects g(1,4,2) in an odd or even number of points. Let Q(double dagger)(x(1),..., x(5)) denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X subset of PG(9, 2) by Y = {y is an element of PG(n,2)\Q(double dagger)(x(1), x(2), x(3), x(4), y) = 0, for all x(1), x(2), x(3), x(4) is an element of X}. Because Q(double dagger) is quinquelinear, the associate X-# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X subset of or equal to X-# while if X is an even 4-flat then X-# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X-# = X. An example of an even 4-flat X such that (X-#)(#) = X is provided by any 4-flat X which is external to g(1,4,2). However, it appears that the two possibilities just illustrated, namely X-# = X for an odd 4-flat and (X-#)(#) = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X-# = PG(9,2) and of even 4-flats for which X-### = X.

Citation

Shaw, R., & Gordon, N. (2004). The quintic Grassmannian g(1,4,2) in PG(9,2). Designs, codes, and cryptography, 32(1-3), 381 - 396. https://doi.org/10.1023/B%3ADESI.0000029236.10701.61

• Journal Article Type: Article
• Acceptance Date: Jul 31, 2004
• Publication Date: Jul 31, 2004
• Journal: DESIGNS CODES AND CRYPTOGRAPHY
• Print ISSN: 0925-1022
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 32
• Issue: 1-3
• Pages: 381 - 396
• DOI: https://doi.org/10.1023/B%3ADESI.0000029236.10701.61
• Keywords: PG(n,2) ; Partial spreads; Grassmannian G1,4,2
• Public URL: https://hull-repository.worktribe.com/output/391639
• Publisher URL: https://link.springer.com/article/10.1023/B%3ADESI.0000029236.10701.61#aboutcontent