On the existence of (v, k, t) trades

A (v, k, t) trade can be used to construct new designs with various support sizes from a given t-design. H.L. Hwang (1986) showed the existence of (v, k, t) trades of volume 2t and the non-existence of trades of volumes less than 2t or of volume 2t + 1. In thIs paper, first we show that there exist (v, k, t) trades of volumes 2t + 2t1 (t ~ 1), 2t + 2t1 + 2t 2 (t ~ 2), 2t + 2t 1 + 2t 2 + 2t 3 (t ~ 3), and 2t+l. Then we prove that, given integers v > k > t ~ 1, there does not exist a (v, k, t) trade of volume s, where 2t < s < 2t + 2t-l.