Pseudorandom generators for combinatorial checkerboards


We define a combinatorial checkerboard to be a function f: {1,., m}d → {1,-1} of the form f(u1.ud)=∏{i=1}dfi(ui) for some functions f i: {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 ε-fools all combinatorial checkerboards with seed length O(log m+log d.log log d+log3/2 1/ε. Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length O(log m+log2d+log d.log 1/ε. Our seed length is better except when 1/ε ≥ dω(log d). © 2012 Springer Basel AG.

Publication Title

Computational Complexity