t-designs on hypergraphs

A $\text{t-}\text{design}\text{T=(X,}\text{B}\text{)}$, denoted by (λ; t, k, v), is a system $\text{B}$ of subsets of size k from a v-set X, such that each t-subset of X is contained in exactly λ elements of $\text{B}$. A hypergraph $\text{H=(Y,}\text{E}\text{)}$ is a finite set Y where $\text{E}\text{=(E}{}_{\mathrm{i}}\text{: i?I)}$ is a family of subsets (which we assume here are distinct) of Y such that E_i≠Ø, i?l, and ?E_i=Y. Let G be an automorphism group of $\text{H=(Y,}\text{E}\text{)}$ where O^l_i is the ith orbit of l-subsets of $\text{E}$. Let A(G; H; t, k)= (a_ij) be an m by n matrix, where a_ij is the number of copies of O^t_i that occur in the system of all t-subsets of all elements of O^k_j. Then there is a t-design $\text{T=(X,}\text{B}\text{)}$ with $\text{X=}\text{E}$, 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).