Binary matroid

In matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2). That is, up to isomorphism, they are the matroids whose elements are the columns of a (0,1)-matrix and whose sets of elements are independent if and only if the corresponding columns are linearly independent in GF(2).

Alternative characterizations
A matroid $$M$$ is binary if and only if
 * It is the matroid defined from a symmetric (0,1)-matrix.
 * For every set $$\mathcal{S}$$ of circuits of the matroid, the symmetric difference of the circuits in $$\mathcal{S}$$ can be represented as a disjoint union of circuits.
 * For every pair of circuits of the matroid, their symmetric difference contains another circuit.
 * For every pair $$C,D$$ where $$C$$ is a circuit of $$M$$ and $$D$$ is a circuit of the dual matroid of $$M$$, $$|C\cap D|$$ is an even number.
 * For every pair $$B,C$$ where $$B$$ is a basis of $$M$$ and $$C$$ is a circuit of $$M$$, $$C$$ is the symmetric difference of the fundamental circuits induced in $$B$$ by the elements of $$C\setminus B$$.
 * No matroid minor of $$M$$ is the uniform matroid $$U{}^2_4$$, the four-point line.
 * In the geometric lattice associated to the matroid, every interval of height two has at most five elements.

Related matroids
Every regular matroid, and every graphic matroid, is binary. A binary matroid is regular if and only if it does not contain the Fano plane (a seven-element non-regular binary matroid) or its dual as a minor. A binary matroid is graphic if and only if its minors do not include the dual of the graphic matroid of $$K_5$$ nor of $$K_{3,3}$$. If every circuit of a binary matroid has odd cardinality, then its circuits must all be disjoint from each other; in this case, it may be represented as the graphic matroid of a cactus graph.

Additional properties
If $$M$$ is a binary matroid, then so is its dual, and so is every minor of $$M$$. Additionally, the direct sum of binary matroids is binary.

define a bipartite matroid to be a matroid in which every circuit has even cardinality, and an Eulerian matroid to be a matroid in which the elements can be partitioned into disjoint circuits. Within the class of graphic matroids, these two properties describe the matroids of bipartite graphs and Eulerian graphs (not-necessarily-connected graphs in which all vertices have even degree), respectively. For planar graphs (and therefore also for the graphic matroids of planar graphs) these two properties are dual: a planar graph or its matroid is bipartite if and only if its dual is Eulerian. The same is true for binary matroids. However, there exist non-binary matroids for which this duality breaks down.

Any algorithm that tests whether a given matroid is binary, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.