# isBinary -- whether a matroid is representable over F_2

## Synopsis

• Usage:
isBinary M
• Inputs:
• M, ,
• Outputs:
• , whether M is binary

## Description

Determines if M is a binary matroid, i.e. is representable over the field $F_2$ of 2 elements.

A matroid is representable over F_2 iff it does not have U_{2,4} as a minor. However, this method does not go through hasMinor, for efficiency reasons: rather it checks whether the symmetric difference of any 2 distinct circuits is dependent.

Note: in general, determining representability is a difficult computational problem. For instance, assuming access to an independence oracle, it is known that the problem of determining whether a matroid is binary cannot be solved in polynomial time.

 i1 : M5 = matroid completeGraph 5 o1 = a matroid of rank 4 on 10 elements o1 : Matroid i2 : isBinary M5 o2 = true i3 : U48 = uniformMatroid(4, 8) o3 = a matroid of rank 4 on 8 elements o3 : Matroid i4 : isBinary U48 o4 = false