On minimum possible volumes of strong Steiner trades

In this note we investigate the minimum possible volumes for strong Steiner trades (SST). We prove that a (v, q + 1,2) SST must have at least q2 blocks if q is even and q2 + q blocks if q is odd. We construct a (v, q+ 1, 2) SST of volume q2 for every q a power of two, and a (v, q+ 1, 2) SST of volume q2 + q, for every q such that q + 1 is a power of two. A construction of (q2 + q + 1, q + 1,2) SSTs of volume q2 + q + 1 is also given for every prime power q. Combinations of these constructions are then used to construct further SSTs. We also show that when the bound for q even is achieved the elements of the trade are the duals of affine planes.