On minimal non-orientable matroids with 2n2n elements and rank nn

We describe an infinite family Mn,kMn,k, with n≥4n≥4 and 1≤k≤n−21≤k≤n−2, of minimal non-orientable matroids of rank nn on a set with 2n2n elements. For k=1,n−2k=1,n−2, Mn,kMn,k is isomorphic to the Bland–Las Vergnas matroid MnMn. For every 2≤k≤n−32≤k≤n−3 a new minimal non-orientable matroid is obtained. All proper minors of the matroids Mn,kMn,k are representable over QQ.