Pseudorandom generators for combinatorial checkerboards

We define a combinatorial checkerboard to be a function f : {1, . . . , m}d → {1,−1} of the form $${f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)}$$ for some functions fi : {1, . . . , m} → {1,−1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1,−1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case m = 2. We construct a pseudorandom generator that $${\epsilon}$$-fools all combinatorial checkerboards with seed length $${O\bigl(\log m+\log d\cdot\log\log d+\log^{3/2} \frac{1}{\epsilon}\bigr)}$$. Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length $${O\bigl(\log m+\log^2d+\log d\cdot\log\frac{1}{\epsilon}\bigr)}$$. Our seed length is better except when $${\frac{1}{\epsilon}\geq d^{\omega(\log d)}}$$.