Ranks of Incidence Matrices of t-designs Sλ(t, t+1, v)

Let S be an Sλ(t, t+1, v) with v>t+ 1 and let Rkp(S) be the rank over GF(p) of an incidence matrix of S; we prove that Rkp(S)=vifp∤t+1, and Rkp(S)=v−1ifp|t+1, except possibly when t = 2 and p = 2 or 3, or when t = 3 and p = 2