T-designs on Hypergraphs

A t-design T=(X, B), denoted by (@l; 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 @l elements of B. A hypergraph H=(Y, E) is a finite set Y where E=(E"i: [email protected]) is a family of subsets (which we assume here are distinct) of Y such that E"i O, [email protected], and @?E"i=Y. Let G be an automorphism group of H=(Y, E) where O^l"i is the ith orbit of l-subsets of E. Let A(G; H; t, k)= (a"i"j) be an m by n matrix, where a"i"j 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 T=(X, B) with X=E, with parameters (@l; 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 @l. The calculus for applying this theorem is illustrated and numerous t-designs for 10=