I am wondering what is the (computationally) best way to **tell if a matroid of size $n$ and rank $r$ is binary(or whether it has a $U^2_4$ minor)** given either one of these:
1) An independence oracle
2) Its rank vector
3) A basis oracle

I want to implement $U^2_4$/binary detection(Most likely given a basis oracle which will be in form of the reverse-lexicographic encoding in [3]).

**What I know(or what I think I know):**

1) can't be solved by asking polynomial number of questions to the oracle which is due to Seymour[1]. 

2) and 3) One can write a brute force implementation which would compute and check all minors of size 4. 

*Also, I considered following ideas:*

**a)** For problem (3), given a matroid $M$ one can start with an $r\times r$ Identity matrix and assign its columns to elements of some basis say $B^*$. Now we expand this matrix by adding columns that correspond to $e\in E(M)\setminus B^*$. We now fill r entries in this new column with $0$ or $1$. For filling $i^{th}$ entry in the column, we ask the basis oracle if $(B^*\setminus i)\cup e$ is a basis. If it is, we put a $1$ in that entry, $0$ otherwise. When we are done constructing all $n-r$ columns this way we have the only binary matrix that correctly reflects if an $r$-subset S of $E(M)$ with $|S\cap B^*|=r-1$ is a basis or not.  As for other $r$-subsets, we can test their rank in the matrix and ask basis oracle whether each of them is a basis. If the two results don't match for any $r$-subset, we conclude that the matroid is not binary representable. This way seems better than the brute-force 'compute and check all minors' in terms of complexity.   

**b)** One can compute all $r-1$ flats of $M$ from bases(Using an algorithm again due to Seymour [2]). Then compute all $r-2$ flats. Then use scum theorem(i.e. check if any $r-2$ flat is contained in more than three $r-1$ flats). 


[1] Seymour, P. D.; Walton, P. N. (1981), "Detecting matroid minors", Journal of the London Mathematical Society, Second Series 23 (2): 193–203

[2] P.D. Seymour, A Note on Hyperplane Generation, Journal of Combinatorial Theory, Series B, Volume 61, Issue 1, May 1994, Pages 88-91 

[3] Yoshitake Matsumoto, Sonoko Moriyama, Hiroshi Imai, and David Bremner. Matroid enumeration for incidence geometry. Discrete & Computational Geometry, 47(1):17–43, 2012