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 dif ficult , 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) .