In this paper we consider a variety of questions in the context of Boolean designs. For example, Erdös asked: How many subsets of an n-set can be found so that pairwise their intersections are all even (odd)? E. Berlekamp  and the author both answered this question; the answer is approximately 2[ 1 2n]. Another question which can be formulated in terms of Boolean designs was asked by J. A. Bondy and D. J. A. Welsh . For what values of d can one find a connected binary matroid of rank d which is identically self-dual? We prove that such matroids exist for all d except 2, 3, and 5. The paper ends with a discussion of more general modular designs and with constructions of some identically self-dual matroids representable over the field of three elements.
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics