A New Infinite Family of Hemisystems of the Hermitian Surface

Abstract

In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q 2 ). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q 2 ) admitting C (q 3 +1)/4 : C 3 .2010 Mathematics Subject Classification. 05B25 (primary), 05E30, 51E12 (secondary).

Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

“…This lemma is a common generalization of the results in [4] and [14]. Its proof is the same as those in [4] and [14]. We therefore omit the proof.…”

“…In a couple of recent papers [7,4], motivated by existence questions concerning finite geometric objects such as m-ovoids and i-tight sets, we used a certain partition of the Singer difference set to construct strongly regular Cayley graphs with special properties which give the desired m-ovoids and i-tight sets. We now realize that the constructions can be done in a more general setting, namely, we can do the construction by partitioning a subdifference set of the Singer difference set in a certain way.…”

“…The second lifting construction (Proposition 3.4) is of "hyperbolic" type, and this constrution is new. In Section 4, we generalize and unify the constructions of strongly regular Cayley graphs corresponding to m-ovoids and i-tight sets in [7,4]. We give a general construction of strongly regular Cayley graphs by using a certain partition of a subdifference set (and its complement) of the Singer difference set.…”

9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

“…(ii) We can apply Theorem 4.11 to the 8th srg in Table 1 as (ℓ, m, p 1 , p 2 , p, e) = (2,1,19,7,5,6). In this case, there exists an integer s 2 such that p s 2 ≡ −1 (mod p 2 ).…”

“…Such a set of lines in H(3, q 2 ) is called a hemisystem, which was first studied by Segre [91]. Constructions of hemisystems can be found in [4,7,29,63].…”