A construction for binary matroids

A family of subsets of a ground set closed under the operation of taking symmetric differences is the family of cycles of a binary matroid. Its circuits are the minimal members of this collection. We use this basic property to derive binary matroids from binary matroids. In particular, we derive two matroids from graphic and cographic matroids. Cocycles of the first one are cutsets or balancing sets. Cocycles of the second one are Eulerian subgraphs or T-joins. We study the problem of finding a minimum weight circuit and cocircuit in these matroids. © 1987.