3‐Designs of PSL(2, 2ⁿ) With block sizes 4 and 5

Abstract

Abstract:We determine the distribution of 3-designs among the orbits of 4-and 5-element subsets under the action of PSLð2; 2 n Þ on the projective line. Thus we give complete information on all Kramer-Mesner matrices for the action of PSLð2; 2 n Þ on 3-sets versus 4-and 5-sets. As a consequence, all 3-designs with block sizes 4 and 5 and automorphism group PSLð2; 2 n Þ can immediately be obtained. #

3-Spontaneous Emission Error Designs from PSL(2, q)or PGL(2, q)

“…Observe that, for some small cases of k, several authors have determined the number of k-element orbits under the action of PGL(2, q) or PSL (2, q) in the process of studying the distribution of 3-designs. These cases correspond to k = 4, 5, 6 in Theorem 3.2 by Cusack et al [10] and Omidi et al [22]; k = 4, 5, 6, 7 in Theorem 3.3 by Keranen et al [15] and Li et al [19,20]; k = 4, 5 in Theorem 3.4 by van Leijenhorst [26]. In particular, Cameron et al [7,8] considered all the cases of k ≡ 0, 1 (mod p) and p n ≡ 1, 3 (mod 4).…”

Simple 3‐designs of PSL(2, 2n) with block size 7

“…The following lemma established in [6] will be used repeatedly. Lemma 2.2 [6]. Let χ k (g) be the number of k-subsets of X fixed by g ∈ G, then χ 1 Proof.…”

“…From Lemma 2.1, we can see that, in order to determine the value of λ, we need only to discuss the order of G B . The following lemma established in [6] will be used repeatedly.…”

“…When q ≡ 3 (mod 4), the 3-designs with block-sizes 4, 5, and 6 with LF(2, q) as an automorphism group are studied in [2] and [10], respectively, and when q ≡ 1 (mod 4), quadruple systems from LF(2, q) are reported in [5]. When q = 2 n , 3-designs with block-sizes 4, 5, and 6 with LF(2, q) as an automorphism group are reported in [6] and [7], respectively. In this article, we consider the case q = 2 n and k = 7.…”

“…Let G denote LF(2, 2 n ) in this paper. The subgroup structure of G is known in [3] and in particular the permutation character χ for this action of G on X is given in Table I [6] below:…”