AbstractA t-design T=(X, B), denoted by (λ; t, k, v), is a system B of subsets of size k from a v-set X, such that each t-subset of X is contained in exactly λ elements of B. A hypergraph H=(Y, E) is a finite set Y where E=(Ei: iϵI) is a family of subsets (which we assume here are distinct) of Y such that Ei≠Ø, iϵl, and ⋃Ei=Y. Let G be an automorphism group of H=(Y, E) where Oli is the ith orbit of l-subsets of E. Let A(G; H; t, k)= (aij) be an m by n matrix, where aij is the number of copies of Oti that occur in the system of all t-subsets of all elements of Okj. Then there is a t-design T=(X, B) with X=E, with parameters (λ; t, k, v), and with G an automorphism groupof T iff there is an m by s submatrix M of A(G; H; t, k) where M has uniform row sums λ. The calculus for applying this theorem is illustrated and numerous t-designs for 10⩽v⩽16 are found and presented. Using a theorem of Alltop on our (12; 4, 6, 13) and (60; 4, 7, 15) we obtain a (12; 5, 7, 14) and a (60; 5, 8, 16).