A family of rank $4$ non-algebraic matroids with pseudomodular dual

The Tic-Tac-Toe matroid is a paving matroid of rank $5$ on 9 elements which is pseudomodular and whose dual is non-algebraic. It has been proposed as a possible example of an algebraic matroid whose dual is not algebraic. We present an infinite family of matroids sharing these properties and generalizing the Tic-Tac-Toe matroid.