On a class of small 2-designs over gf(q)

A 2 ? (v,k,λ;q) design is a pair (V, B) of a v-dimensional vector space V over GF(q) and a collection B of k-dimensional subspaces of V such that each 2-dimensional subspace of V is contained in exactly λ members of B. Assuming transitivity of their automorphism groups on the nonzero vectors of V, we give a classification of nontrivial such designs for v = 7, q = 2,3 with small λ, together with the nonexistence proof of those designs for v ? 6. © 1995 John Wiley & Sons, Inc.