Delandtsheer–Doyen parameters for block-transitive point-imprimitive 2-designs

Abstract

Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters m, n, now called Delandtsheer–Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer–Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of designs achieving these bounds, and pose several open questions concerning the Delandtsheer–Doyen parameters.

Groups of prime degree and the Bateman-Horn Conjecture

“…In the corrigendum of [12] it is shown that if G is primitive and the socle M of G/Z is imprimitive and non-abelian, then the commutator subgroup G ′ is imprimitive and isomorphic to PSL n (q) where m = (q n − 1)/(q − 1), with q odd or G ′ PSL 3 (2) if n ≥ 3. Conversely, such groups G exist provided m ≥ 5.…”

“…Block designs. A construction by Amarra, Devillers and Praeger [2] of block-transitive pointimprimitive 2-designs with specific parameters depends on certain polynomials, such as f (t) = 32t 2 +20t+1, taking prime power values. Using all primes r < 10 8 gives C( f ) = 4.721240, and Li's modified BHC then gives an estimate E(10 8 ) = 12 362 961.06.…”

“…The Goormaghtigh Conjecture (1917). Since 1 + 2 + 2 2 + 2 3 + 2 4 = 31 = 1 + 5 + 5 2 , both PSL 5 (2) and PSL 3 (5) have natural degree 31. Goormaghtigh, a Belgian engineer and amateur mathematician, conjectured in [21] that this example and 1 + 2 + 2 2 + • • • + 2 12 = 8191 = 1 + 90 + 90 2 are the only positive integer solutions of (x n − 1)/(x − 1) = (y k − 1)/(y − 1) with n k and n, k ≥ 3.…”

Block designs and prime values of polynomials

“…Block designs play a major role in combinatorics and finite geometry, and have many applications in statistics, specifically in the design of experiments. In [2], Amarra, Devillers and Praeger have recently constructed families of highly symmetric 2-designs which maximise certain parameters. Their constructions depend on certain quadratic polynomials with integer coefficients taking prime power values.…”

“…Nevertheless, the Bateman-Horn Conjecture [4], dating from 1962 and also proved only for degree 1, gives estimates E(x) for the number Q(x) of positive integers t ≤ x at which a given polynomial takes prime values. Using a recent improvement to the Bateman-Horn Conjecture due to Li [22], we calculated these estimates E(x) for some of the simpler polynomials arising in [2], taking x = 10 8 , and compared them with the actual numbers Q(x) found by computer searches. As in various other applications of this conjecture (see [19,20] for example), the estimates E(x) are remarkably close to the actual values Q(x).…”