Base matrices of various heights

Abstract A classical theorem of Balcar, Pelant, and Simon says that there is a base matrix height ${\mathfrak h}$ , where the distributivity number ${\cal P} (\omega ) / {\mathrm {fin}}$ . We show if continuum c}$ regular, then are matrices any regular uncountable $\leq {\mathfrak in Cohen random models. This answers questions Fischer, Koelbing, Wohofsky.