A formula for the number of Steiner quadruple systems on 2 n points of 2‐rank 2 n −n

Assmus [1] gave a description of the binary code spanned by the blocks of a Steiner triple or quadruple system according to the 2‐rank of the incidence matrix. Using this description, the author [13] found a formula for the total number of distinct Steiner triple systems on 2 n −1 points of 2‐rank 2 n ‐ n . In this paper, a similar formula is found for the number of Steiner quadruple systems on 2 n points of 2‐rank 2 n ‐ n . The formula can be used for deriving bounds on the number of pairwise non‐isomorphic systems for large n , and for the classification of all non‐isomorphic systems of small orders. The formula implies that the number of non‐isomorphic Steiner quadruple systems on 2 n points of 2‐rank 2 n ‐ n grows exponentially. As an application, the Steiner quadruple systems on 16 points of 2‐rank 12 are classified up to isomorphism. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 260–274, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10036