TY - JOUR

T1 - Boolean designs and self-dual matroids

AU - Graver, Jack E.

N1 - Funding Information:
*The research was supported in part by the National Science Foundation, GP-19404, and in part by the U.S. Army, DAJA37-72-C-1519, (while the author was visiting Nottingham University).

PY - 1975/4

Y1 - 1975/4

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.

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

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

M3 - Article

AN - SCOPUS:0016497311

SN - 0024-3795

VL - 10

SP - 111

EP - 128

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 2

ER -