A new infinite family of minimal non-orientable matroids of rank 3 with 3n elements

We construct a new infinite family of minimal non-orientable matroids of rank 3 with 3n elements starting from . Two minimal non-orientable matroids of rank 3 with 9 elements, and , were discovered in 2012. We then consider the representability of and ; in particular the new infinite family is not representable over fields with characteristic 2.