Designs and Codes from Basic 3-Transpositions

In this paper, we aim to study maximal pairwise commuting sets of 3-transpositions, and to construct designs and codes from these sets. Any maximal set of pairwise 3-transpositions is called a basic set of transpositions. Let G be a 3-transposition group with the set D as the conjugacy class consisting of its 3-transpositions. Let L be a set of basic transpositions in D. We aim to give a general description of 1-(v, k, λ) designs D = (P, B), with P = D and B = {Lg|g ∈ G}. The parameters k = |L|, λ and further properties of D are determined. In addition, some of the codes associated with these designs are also discussed.We also, as examples, apply the method to Symmetric groups Sn and Fischer groups Fi for i ∈ {21, 22, 23, 24}.