# Classification of flocks of the quadratic cone over fields of order at most 29

@article{Law2003ClassificationOF, title={Classification of flocks of the quadratic cone over fields of order at most 29}, author={Maska Law and Tim Penttila}, journal={Advances in Geometry}, year={2003}, volume={2003} }

We complete the classification of flocks of the quadratic cone in PGð3; qÞ for qc 29, by showing by computer that there are exactly 8 flocks of the quadratic cone in PGð3; 19Þ ,1 8 flocks of the quadratic cone in PGð3; 23Þ, 12 flocks of the quadratic cone in PGð3; 25Þ ,1 4 flocks of the quadratic cone in PGð3; 27Þ, and 28 flocks of the quadratic cone in PGð3; 29Þ ,u p to equivalence.

## 10 Citations

Construction of BLT-sets over small fields

- Mathematics, Computer ScienceEur. J. Comb.
- 2004

The computer algebra system MAGMA is used to search for BLT-sets of the nonsingular parabolic quadric Q(4, q), giving rise to 158 new flocks of the quadratic cone in PG(3, q).

Rainbow cliques and the classification of small BLT-sets

- Mathematics, Computer ScienceISSAC '13
- 2013

The classification problem of these sets is reduced to the problem of classifying rainbow cliques in graphs, which allows us to classify BLT-sets for all orders q in the range 31 to 67.

Classification of 8-dimensional rank two commutative semifields

- Mathematics
- 2016

Abstract We classify the rank two commutative semifields which are 8-dimensional over their center 𝔽q. This is done using computational methods utilizing the connection to linear sets in PG(2, q4).…

ELATION GENERALIZED QUADRANGLES FOR WHICH THE NUMBER OF LINES ON A POINT IS THE SUCCESSOR OF A PRIME

- Journal of the Australian Mathematical Society
- 2008

Abstract We show that an elation generalized quadrangle that has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (that is, is a flock quadrangle).

Elation generalised quadrangles of order (s, p), where p is prime

- Mathematics
- 2007

We show that an elation generalised quadrangle which has p + 1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle).

Hemisystems of small flock generalized quadrangles

- Mathematics, Computer ScienceDes. Codes Cryptogr.
- 2013

By analysing the computational data, this paper identifies two possible new infinite families of hemisystems in the classical generalized quadrangle H(3, s2).

A hemisystem of a nonclassical generalised quadrangle

- Mathematics, Computer ScienceDes. Codes Cryptogr.
- 2009

It is shown that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 52), which leads to a new partial quadrangles and hence a strongly regular point graph.

A Class of Transitive BLT-Sets

- Mathematics
- 2010

We present a class of transitive BLT-sets that contains several known examples and at least one new example over the field with elements with automorphism group . This example is sporadic in the…

A Rainbow Clique Search Algorithm for BLT-Sets

- Computer ScienceICMS
- 2018

An algorithm to search for rainbow cliques in vertex-colored graphs is discussed and a larger algorithm to classify a certain type of geometric-combinatorial objects called BLT-sets is described.

Every flock generalized quadrangle has a hemisystem

- Mathematics
- 2010

We prove that every flock generalised quadrangle contains a hemisystem, and we provide a construction method which unifies our results with the examples of Cossidente and Penttila in the classical…

## References

SHOWING 1-10 OF 74 REFERENCES

Flocks of a quadratic cone in PG(3, q), q ≤ 8

- Mathematics
- 1988

If ℒ(O) is a quadratic cone in PG(3,q), with vertex x, then a flock of ℒ(O) is a partition of ℒ(O)-{x} into q disjoint conics. With such a flock there correspond a translation plane of order q2 and a…

Generalized Quadrangles and Flocks of Cones

- Computer Science, MathematicsEur. J. Comb.
- 1987

It is proved 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) and with each flock of the quadratic cone there corresponds such a setof matrices.

On the flocks of Q+(3,q)

- Mathematics
- 1989

A complete characterization of the flocks of Q+(3, q) is given. As an application, it follows that if q is odd, q≠11, 23, 59, there exist no maximal exterior sets of Q+ (2n−1, q) for n>2.

Flocks of hyperbolic quadrics and linear groups containing homologies

- Mathematics
- 1992

The classification of the subgroups of PGL(2,q) can be used to obtain the classification of the flocks of the hyperbolic quadrics in PG(3,q). A description of the group of all linear collineations…

Conical flocks, partial flocks, derivation, and generalized quadrangles

- Mathematics
- 1991

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.…

Flocks and ovals

- Mathematics
- 1996

An infinite family of q-clans, called the Subiaco q-clans, is constructed for q=2e. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q2, q),…

A unified construction of finite geometries associated with q-clans in characteristic 2

- Mathematics
- 2003

Flocks of Laguerre planes, generalized quadrangles, translation planes, ovals, BLT- sets, and the deep connections between them, are at the core of a developing theory in the area of geometry over…

Construction of BLT-sets over small fields

- Mathematics, Computer ScienceEur. J. Comb.
- 2004

The computer algebra system MAGMA is used to search for BLT-sets of the nonsingular parabolic quadric Q(4, q), giving rise to 158 new flocks of the quadratic cone in PG(3, q).

The Law—Penttila q -Clan Geometries

- Mathematics
- 2001

For a prime power q, a q-clan is a set of q 2 × 2 matrices over GF(q) whose pairwise differences give anisotropic quadratic forms. These objects are used to construct translation planes of order q 2,…

BLT-sets over small fields

- Mathematics, Computer ScienceAustralas. J Comb.
- 1998

The relationship between BLT-sets and flocks of the quadratic cone in PG(3, q) means that this work classifies flocks for q ::; 17 and finds new flock for q = 19, q = 23 and q = 25, which yield 26 new flocks.