Non-full-rank Steiner quadruple systems S(v, 4, 3)

Abstract

All different Steiner systems S(2 m , 4, 3) of order 2 m and rank 2 m ? m ? 1 + s over $$\mathbb{F}_2$$ , where 0 ≤ s ≤ m ? 1, are constructed. The number of different systems S(2 m , 4, 3) whose incident matrices are orthogonal to a fixed code is obtained. A connection between the number of different Steiner systems and of different Latin cubes is described.