### Abstract

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 [2] 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 [1]. 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.

Original language | English (US) |
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Pages (from-to) | 111-128 |

Number of pages | 18 |

Journal | Linear Algebra and Its Applications |

Volume | 10 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1975 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics