Trades and t-designs
G. B. Khosrovshahi, Behruz Tayfeh‐Rezaie · Cambridge University Press eBooks · 2009
Trades, as combinatorial objects, possess interesting combinatorial and algebraic properties and play a considerable role in various areas of combinatorial designs. In this paper we focus on trades within the context of t -designs. A pedagogical review of the applications of trades in constructing halving t -designs is presented. We also consider ( N, t )-partitionable sets as a generalization of trades. This generalized notion provides a powerful approach to the construction of large sets of t -designs. We review the main recursive constructions and theorems obtained by this approach. Finally, we discuss the linear algebraic representation of trades and present two applications. Introduction Let v, k, t and λ be integers such that v ≥ k ≥ t ≥ 0 and λ ≥ 1. Let X be a v -set and let P i ( X ) denote the set of all i -subsets of X for any i . A t -( v, k, λ ) design (briefly t-design ) is a pair D =( X, B )inwhich B is a collection of elements of P k ( X ) such that every A ∈ P t ( X ) appears in exactly λ elements of B . Let N be a natural number greater than 1.