On inequivalent representations of matroids over non-prime fields

For each finite field F of prime order there is a constant c such that every 4-connected matroid has at most c inequivalent representations over F . We had hoped that this would extend to all finite fields, however, it was not to be. The ( m , n ) -mace is the matroid obtained by adding a point freely to M ( K m , n ) . For all n ⩾ 3 , the ( 3 , n ) -mace is 4-connected and has at least 2 n representations over any field F of non-prime order q ⩾ 9 . More generally, for n ⩾ m , the ( m , n ) -mace is vertically ( m + 1 ) -connected and has at least 2 n inequivalent representations over any finite field of non-prime order q ⩾ m m .