Distance-Regular Graphs

Finite Generalized Quadrangles

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

/pdf/the-geometry-of-two-weight-codes-nnndeshbae.pdf

The Geometry of Two-Weight Codes

https://jupiter.math.nctu.edu.tw/~weng/references/distance-regular graphs/cospec_2003.pdf?origin=publication_detail

Which graphs are determined by their spectrum

Terminology Quadrics Hermitian varieties Grassmann varieties Veronese and Segre varieties Embedded geometries Arcs and caps Appendix VI. Ovoids and spreads of finite classical polar spaces Appendix VII. Errata for Finite projective spaces of three dimensions and Projective geometries over finite fields Bibliography Index of notation Author index General index.

/pdf/general-galois-geometries-15ofausktq.pdf

General Galois geometries

A flock of the quadratic cone K of PG(3, q) is a partition of K but its vertex into disjoint conics. If the planes of the q conics of such a flock all contain a common line, then the flock is called linear. With any flock there corresponds a translation plane which is Desarguesian iff the flock is linear. W. M. Kantor showed that with a set of q upper triangular 2 × 2-matrices over GF(q) of a certain type, there corresponds a generalized quadrangle of order (q2, q). We prove that with such a set of q matrices there corresponds a flock of the quadratic cone of PG(3, q), and conversely that with each flock of the quadratic cone there corresponds such a set of matrices. Using this relationship, new flocks, new generalized quadrangles, and probably new translation planes are obtained.

Generalized quadrangles and flocks of cones

LetP be a finite classical polar space of rankr, r⩾2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceW

 n
 
(q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q−(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q
2) arising from a non-singular Hermitian variety inPG(n, q
2)n even andn > 2, have no ovoids. LetS be a generalized hexagon of ordern (⩾1). IfV is a pointset of order n3 + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG
2
(q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=32h+1, has an ovoid, and thatH(q), q even, has no ovoid. A regular system of orderm onH(3,q
2) is a subsetK of the lineset ofH(3,q
2), such that through every point ofH(3,q
2) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3).

Ovoids and spreads of finite classical polar spaces

Publisher Summary
This chapter focuses on projective geometry over a finite field. A k-arc in projective plane, PG (n, q) is a set K of k points with k ≥ n + 1 such that no n + 1 points of K lie in a hyperplane. An arc K is complete if it is not properly contained in a larger arc. A normal rational curve of PG(2, q) is an irreducible conic; a normal rational curve of PG(3, q) is a twisted cubic. It is well known that any (n + 3)-arc of PG(n, q) is contained in a unique normal rational curve of this space. For q > n + 1, the osculating hyperplane of the normal rational curve C at the point x ϵ C is the unique hyperplane through x intersecting C at x with multiplicity n. In PG(n, q), n ≥ 3, a set K of k points no three of which are collinear is a k-cap. A k-cap is complete if it is not contained in a (k + 1)-cap. A line of PG(n, q) is a secant, a tangent, or an external line of a k-cap as it meets K in 2, 1, or 0 points.

Joseph A. Thas

Papers

Finite Generalized Quadrangles

General Galois geometries

Generalized quadrangles and flocks of cones

Ovoids and spreads of finite classical polar spaces

Projective Geometry over a Finite Field