The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Abstract

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v ≡ 39 (mod 72), or $$v \equiv 4^e48 + 3$$ v ≡ 4 e 48 + 3 (mod $$4^e96$$ 4 e 96 ) and $$e \ge 0$$ e ≥ 0 , there exists a KTS on v points having at least $$v-3$$ v - 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

On optimal (Z6m×Z6n,4,1) and (Z2m×Z18n,4,1) difference packings and their related codes

“…Proof. By Remark of Theorem 3.14, there is a m n s t (6 × 6 , ′ × ′, 4, 1)-RDF where s t ( ′, ′) {(2, 6), (6, 2), (6,6), (2,18), (18, 2)} ∈ . Applying Lemma 3.7 with the following two DPs, we obtain an optimal m n (6 × 6 , 4, 1)-DP.…”

“…Recently some work has been done on SDFs [16][17][18]32]. And strong difference families have been used to construct 3-pyramidal Kirkman triple systems, partition difference families and relative difference families (see [2,8,11,24]). And the last theorem in [9]…”

“…The blocks of an m n ( × , 3 × 3, 4, 1)-RDF are listed below: m n ( , ) = (9, 9): {(0, 0), (0, 1), (1, 1), (2, 3)}, {(0, 0), (0, 2), (1, 5), (3, 1)}, {(0, 0), (1,4), (5,5), (8, 3)}, {(0, 0), (1, 7), (3, 2), (7, 2)}, {(0, 0), (2, 0), (4, 6), (6, 5)}, {(0, 0), (4, 2), (8, 1), (4, 7)}. m n ( , ) = (3, 27): {(0, 0), (0, 1), (1, 2), (2, 5)}, {(0, 0), (0, 2), (1,6), (2, 13)}, {(0, 0), (0, 3), (2,19), (0, 11)}, {(0, 0), (0, 4), (2, 7), (1, 21)}, {(0, 0), (0, 5), (2,17), (0, 15)}, {(0, 0), (0, 6), (2, 1), (0, 13)}. □ = {(2, 18), (6,6), (6,12), (2, 54), (6,18), (6,24), (6,36), (6,48), ∈ (6, 72), (6,144), (18,12), (18,24), (18, 48)}.…”

“…Assume that there is an m m t ( × , × , 4, 1) (3 × 3, 4, 1)-DM in Lemma 3.1, applying Lemmas 3.6 and 3.7 with the known (6 × 18, 6 × 2, 4, 1)-RDF in Lemma 2.6 gives a (6 × 144, 6 × 2, 4, 1)-RDF and a (18 × 48, 6 × 2, 4, 1)-RDF. Hence, we can obtain a m n s t (6 × 6 , ′ × ′, 4, 1)-RDF for any positive integers m n , , where s t ( ′, ′) {(2, 6), (6, 2), (6,6), (2,18), (18, 2)} ∈ .…”

“…m n ( , ) = (18, 6): {(0, 0), (17,4), (2,4), (3, 2)}, {(0, 0), (16, 0), (4, 1), (5, 2)}, {(0, 0), (2, 2), (7, 0), (10, 5)}, {(0, 0), (2, 5), (7,5), (1, 5)}, {(0, 0), (15, 2), (13, 5), (3, 1)}, {(0, 0), (15,3), (4, 0), (8, 5)}, {(0, 0), (16,5), (4, 2), (12, 2)}, {(0, 0), (17,3), (5,5), (10, 2)}.…”