Abstract

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 Pi(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 Pk (X) such that every A ∈ Pt(X) appears in exactly λ elements of B. Let N be a natural number greater than 1.