Some remarks on flocks

Abstract

New proofs are given of the fundamental results of Bader, Lunardon and Thas relating flocks of the quadratic cone in PG(3, q), q odd, and BLT-sets of Q(4, q). We also show that there is a unique BLT-set of H(3, 9). The model of Penttila for Q(4, q), q odd, is extended to Q(2m, q) to construct partial flocks of size qm/2 + m/2 -1 of the cone Jt in PG(2m -1, q) with vertex a point and base Q(2m -2,q), where q is congruent to 1 or 3 modulo 8 and m is even. These partial flocks are larger than the largest previously known for m > 2. Also, the example of O'Keefe and Thas of a partial flock of Jf in PG(5, 3) of size 6 is generalised to a partial flock of the cone Jt? of PG(2pn -1, p) of size 2pn, for any prime p congruent to 1 or 3 modulo 8, with the corresponding partial BLT-set of Q(2pn, p) admitting the symmetric group of degree 2pn + 1.2000 Mathematics subject classification: primary 51E12; secondary 51A40, 51A5O, 51E20, O5B25.

Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

“…From r q+1 1 +r q+1 2 = 0, it follows that Tr q 2 /q (r 1 x q 1 ) = a. As for any given a there are q elements x ∈ GF(q 2 ) such that Tr q 2 /q (x) = a, we see that p 1 13 = p 1 15 = q(q − 1). Finally p 1 14 = η 1 − (p 1 10 + p 1 11 + p 1 12 + p 1 13 + p 1 15 ) = q(q − 1) 2 .…”

“…for a fixed µ ∈ GF(q 2 ) with N q 2 /q (µ) = −1, and set ρ = τ • κ. It turns out that the lines of H(3, q 2 ) defined by (12) are mapped by ρ to the points of the Q − (5, q) defined by Q in (1). For any point P ∈ H(3, q 2 ), by abuse of notation, we write ρ(P ) to denote the totally singular line {ρ(r) : r is a totally isotropic line on P }.…”

“…From a result by De Soete and Thas [8], q is necessarily odd. Bader, O'Keefe, Penttila in [1] and, independently, Shult in [16] constructed an example of a special set of H(3, q 2 ). This consists of the q 2 + 1 points of an elliptic quadric over GF(q) which is the complete intersection of H(3, q 2 ) with a hyperbolic quadric of PG(3, q 2 ) whose polarity commutes with the given unitary one.…”

Rainbow cliques and the classification of small BLT-sets

“…For the converse, we need to show that a clique C in ΓS of size q + 1 − s defines a BLT-set T := S ∪ C. For this purpose, we need to look at all triples P, Q, R ∈ T. The only interesting case is when P, Q, R ∈ C. In this case, pick a point P0 ∈ S. The presence of the three edges P Q, P R, and QR in ΓS implies that the triples P0P Q, P0P R and P0QR are partial BLT-sets. By [2,Lemma 4.3], the triple P QR is partial BLT, too.…”

Special sets of the Hermitian surface and indicator sets

“…It should be noted that if q = 3 any special set of H (3,9) is a Baer elliptic quadric, [1], and that there exist elliptic quadrics Q − (3, q) embedded in H(3, q 2 ), q odd, that cannot be obtained by means of hyperbolic quadrics commuting with H(3, q 2 ). When q is even, any elliptic quadric Q − (3, q) embedded in H(3, q 2 ) cannot be a special set.…”