### 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) |
---|---|

Pages (from-to) | 111-128 |

Number of pages | 18 |

Journal | Linear Algebra and Its Applications |

Volume | 10 |

Issue number | 2 |

DOIs | |

State | Published - 1975 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

**Boolean designs and self-dual matroids.** / Graver, Jack E.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 10, no. 2, pp. 111-128. https://doi.org/10.1016/0024-3795(75)90003-8

}

TY - JOUR

T1 - Boolean designs and self-dual matroids

AU - Graver, Jack E

PY - 1975

Y1 - 1975

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0016497311&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0016497311&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(75)90003-8

DO - 10.1016/0024-3795(75)90003-8

M3 - Article

AN - SCOPUS:0016497311

VL - 10

SP - 111

EP - 128

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 2

ER -