Some characterizations of quasi-symmetric designs with a spread

We prove that every polarity of PG(2k − 1, q), where k ≥ 2, gives rise to a design with the same parameters and the same intersection numbers as, but not isomorphic to, PG k (2k, q). In particular, the case k = 2 yields a new family of quasi-symmetric designs. We also show that our construction provides an infinite family of counterexamples to Hamada's conjecture, for any field of prime order p. Previously, only a handful of counterexamples were known.

Section: New Quasi-symmetric Designsmentioning

confidence: 91%

Polarities, quasi-symmetric designs, and Hamada’s conjecture

Jungnickel

Tonchev

2008

“…We remark that the special case n = 4 was previously obtained by Sane and Shrikhande [11] under strong additional hypotheses: D was assumed to be quasi-symmetric with the correct intersection numbers 1 and q + 1. By the Dembowski-Wagner theorem (see, for instance, [1, Theorem XII.2.10]), the analogous characterization via line size q + 1 also holds for designs with the parameters of PG n−1 (n, q).…”

Section: More On Hyperplanesmentioning

confidence: 98%

The number of designs with geometric parameters grows exponentially

Jungnickel

Tonchev

2009

It is well-known that the number of 2-designs with the parameters of a classical point-hyperplane design PG n−1 (n, q) grows exponentially. Here we extend this result to the number of 2-designs with the parameters of PG d (n, q), where 2 ≤ d ≤ n − 1. We also establish a characterization of the classical geometric designs in terms of hyperplanes and, in the special case d = 2, also in terms of lines. Finally, we shall discuss some interesting configurations of hyperplanes arising in designs with geometric parameters. Mathematics Subject Classifications (2000) 05B05 · 51E20 PreliminariesIt is well-known that the number of symmetric 2-designs with the parameters of a classical point-hyperplane design PG n−1 (n, q) grows exponentially; a similar result holds for affine 2-designs with the parameters of a classical point-hyperplane design AG n−1 (n, q). These results were originally established by the first author [6], whose bounds were subsequently improved in several papers [8][9][10].In this paper, we shall establish a corresponding result for the number of 2-designs with the parameters of PG d (n, q), where d is in the range 2 ≤ d ≤ n − 2. We also obtain a Communicated by Ron Mullin.To Spyros Magliveras on the occasion of his 70th birthday.

“…As a consequence, one approach in the study of such designs has been to put additional parametric or structural restrictions. Baartmans and Shrikhande [1]; Limaye, Sane, and Shrikhande [11]; Mavron and Shrikhande [13] ; Cameron [6]; Sane and Shrikhande [24]; McDonough and Mavron [16]; Mavron, McDonough and Shrikhande [14] are some papers where additional structural conditions are imposed. Pawale [21] studies quasi-symmetric 2-designs satisfying a parametric condition of the form y − x has a fixed value.…”

mentioning

confidence: 99%

On quasi-symmetric designs with intersection difference three

Mavron

McDonough

Shrikhande

2011

Self Cite

In a recent paper, Pawale (Des Codes Cryptogr, 2010) investigated quasisymmetric 2-(v, k, λ) designs with intersection numbers x > 0 and y = x + 2 with λ > 1 and showed that under these conditions either λ = x + 1 or λ = x + 2, or D is a design with parameters given in the form of an explicit table, or the complement of one of these designs. In this paper, quasi-symmetric designs with y − x = 3 are investigated. It is shown that such a design or its complement has parameter set which is one of finitely many which are listed explicitly or λ ≤ x + 4 or 0 ≤ x ≤ 1 or the pair (λ, x) is one of