Finite Generalized Quadrangles

Determine the value of the variable in each equation.

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

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Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory - developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems. The book is aimed at mathematicians working in either coding theory or combinatorics - or related areas of algebra. The book is, however, designed to be used by non-specialists and can be used by those graduate students or computer scientists who may be working in these areas.

Designs and their codes

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s
2), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3
 e
 ),e≥3, is constructed. Then, by the duality betweenQ(4, 3
 e
 ) and the classical generalized quadrangleW (3
 e
 ), we get line spreads of PG(3, 3
 e
 ) and hence translation planes of order 32e
. These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q
2,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q
2),q even, having a subquadrangleS′ isomorphic toQ(4,q) and if inS′ each ovoid consisting of all points collinear with a given pointx ofS\S′ is an elliptic quadric, thenS is isomorphic toQ(5,q).

Spreads and ovoids in finite generalized quadrangles

We study a class of generalized quadrangles satisfying a local Moufang condition that is more restrictive than the usual concept of skew translation GQ. A new infinite family is constructed whose point-line duals are translation generalized quadrangles with very small kernel. They have parameters (q2, q), q=3r, for r>2.

An essay on skew translation generalized quadrangles

L. Bader, G. Lunardon and J. A. Thas have shown that a flock ℱ0 of a quadratic cone in PG(3, q), q odd, determines a set ℱ={ℱ0,ℱ1,...,ℱq} of q+1 flocks. Each ℱj, 1≦j≦q, is said to be derived from ℱ0. We show that, by derivation, the flocks with q=3e arising from the Ganley planes yield an inequivalent flock for q≧27. Further, we prove that the Fisher flocks (q odd, q≧5) are the unique nonlinear flocks for which (q−1)/2 planes of the flock contain a common line. This result is used to show that each of the flocks derived from a Fisher flock is again a Fisher flock. Finally, we prove that any set of q−1 pairwise disjoint nonsingular conics of a cone can be extended to a flock. All these results have implications for the theory of translation planes.

Stanley E. Payne

Papers

Finite Generalized Quadrangles

Spreads and ovoids in finite generalized quadrangles

An essay on skew translation generalized quadrangles

Conical flocks, partial flocks, derivation, and generalized quadrangles

Nonisomorphic generalized quadrangles