Abstract
The cogirth, g⁎(M), of a matroid M is the size of a smallest cocircuit of M. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that g⁎(M)≤12|E(M)| when M is binary, unless M simplifies to a projective geometry. We also show that, when equality holds, M simplifies to a Bose-Burton geometry, that is, a matroid of the form PG(r−1,2)−PG(k−1,2). These results extend to matroids representable over arbitrary finite fields.
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