T-(ν,k) trades and t-(ν, k) latin trades

Abstract

Let [n] denotes the set {1,2,⋯, n}. A t-(ν k) trade T = (Ti, Ta) is a pair of two disjoint collections of fc-subscts of [ν] (called blocks) such that for every t-subsct of [ν], the number of blocks containing this subset is the same in both T1 and T2. By imposing some order on each block, t-(ν, k) trades may be generalized to t-(ν, k) Latin trades. t-(ν, k) trades arc useful in the study of block designs, while t-(ν, k) Latin trades arc related to Latin squares and orthogonal arrays. Here we show relations between these two combinatorial objects and present some new results on the spectrum (thai is, the set of allowable vol-umes) of t-(ν, k) Latin trades. By this method wc produce some t-(ν, k) trades with previously unknown volumes.