On the existence of d-homogeneous 3-way Steiner trades

Abstract

A μ-way (v, k, t) trade T = {T₁, T₂,-, Tμ} of volume m consists of fj. disjoint collections T₁,T₂,...,T_μ}, each of m blocks of size k, such that for every t-subset of v-set V the number of blocks containing this t-subset is the same in each T₁ (for 1 ≤ i ≤μ). A μ-way (v, k, t) trade is called μ-way (v, k, t) Steiner trade if any t-subset of found(T) occurs at most once in T₁ (T_j, j ≥ 2). A μ-way (v, k, t) trade is called d-homogeneous if each element of V occurs in precisely d blocks of T\ (Tj, j ≥ 2). In this paper we characterize the 3-way 3-homogeneous (v, 3,2) Steiner trades of volume v. Also we show how to construct a 3-way d-homogeneous (v, 3,2) Steiner trade for d {4,5,6}, except for seven small values of v. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.