Another class of balanced graph designs: balanced circuit designs

A block considered as a set of elements together with its adjacency matrix A is called a C-block if A is the adjacency matrix of a circuit. A balanced circuit design with parameters v, b, r, k, λ (briefly, BCD(v, k, λ)) is an arrangement of v elements into b C-blocks such that each C-block contains k elements, each element occurs in exactly r C-blocks and any two distinct elements are linked in exactly λ C-blocks.We investigate conditions for the existence of BCD and show, in particular, that if the block-size k ⩽ 6 and the trivial necessary conditions are satisfied, then the corresponding BCD exists.