On minimal non-orientable matroids with elements and rank

We describe an infinite family M n , k , with n ≥ 4 and 1 ≤ k ≤ n − 2 , of minimal non-orientable matroids of rank n on a set with 2 n elements. For k = 1 , n − 2 , M n , k is isomorphic to the Bland–Las Vergnas matroid M n . For every 2 ≤ k ≤ n − 3 a new minimal non-orientable matroid is obtained. All proper minors of the matroids M n , k are representable over Q .