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

✍ Scribed by Vladimir D. Tonchev

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 133 KB
• Volume: 11
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.10036

✦ Synopsis

Abstract

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