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San Jose, California USA
June 8, 2011 to June 11, 2011
ISBN: 978-0-7695-4411-3
pp: 232-242
ABSTRACT
We define a combinatorial checkerboard to be a function $f:\{1,\ldots,m\}^d\to\{1,-1\}$ of the form $f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)$ for some functions $f_i:\{1,\ldots,m\}\to\{1,-1\}$. 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 $\frac{1}{\epsilon}\geq d^{\omega(\log d)}$.
INDEX TERMS
pseudorandom generators, combinatorial checkerboards
CITATION
Thomas Watson, "Pseudorandom Generators for Combinatorial Checkerboards", CCC, 2011, 2012 IEEE 27th Conference on Computational Complexity, 2012 IEEE 27th Conference on Computational Complexity 2011, pp. 232-242, doi:10.1109/CCC.2011.12
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