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The p-rank of a Steiner triple system B is the dimension of the linear span of the set of characteristic vectors of blocks of B, over GF(p). We derive a formula for the number of different Steiner triple systems of order v and given 2-rank r_2<v and the number of Steiner triple systems of order v and given 3-rank r_3<v-1. We prove that there are no Steiner triple systems of 2-rank smaller than v and, at the same time, 3-rank smaller than v-1.arXiv:1806.00009v1 fatcat:petcjh7mo5cutcgzu7yq6q5n4e