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Subsets of F_2^n that are eps-biased, meaning that the parity of any set ofbits is even or odd with probability eps close to 1/2, are powerful tools forderandomization. A simple randomized construction shows that such sets exist ofsize O(n/eps^2), and known deterministic constructions achieve sets of sizeO(n/eps^3), O(n^2/eps^2), and O((n/eps^2)^{5/4}). Rather than derandomizingthese sets completely in exchange for making them larger, we attempt a partialderandomization while keeping them small, constructing sets of size O(n/eps^2)with as few random bits as possible. The naive randomized construction requiresO(n^2/eps^2) random bits. We give two constructions. The first uses Nisan'sspace-bounded pseudorandom generator to partly derandomize a folkloreprobabilistic construction of an error-correcting code, and requires O(n log(1/eps)) bits. Our second construction requires O(n log (n/eps)) bits, but ismore elementary; it adds randomness to a Legendre symbol construction on Alon,Goldreich, H{\aa}stad, and Peralta, and uses Weil sums to bound high moments ofthe bias.

Optimal epsilon-biased sets with just a little randomness