Consequences of the Brylawski–Lucas Theorem for Binary Matroids

The principal theme of the present paper is to consider isomorphism classes of binary matroids as orbits of a suitable group action. This interpretation is based on a theorem of Brylawski–Lucas. A refinement of the Burnside Lemma is used in order to enumerate these orbits. Ternary matroids are dealt with in much the same way (Section 2). Counting regular matroids is more difficult, but their number can be estimated with an arbitrarily small relative error (Section 3). Other applications of the Brylawski–Lucas Theorem include checking binary matroids for isomorphism (Section 4) and for graphicness (Section 5).