Small-bias sets for nonabelian groups: Derandomizations of the Alon-Roichman theorem

Abstract

In analogy with ε-biased sets over ℤ2n, we construct explicit ε-biased sets over nonabelian finite groups G. That is, we find sets S ⊂ G such that ∥double-struck Ex∈S ρ(x)∥ ≤ ε for any nontrivial irreducible representation ρ. Equivalently, such sets make G's Cayley graph an expander with eigenvalue |λ| ≤ ε. The Alon-Roichman theorem shows that random sets of size O(log|G|/ε2) suffice. For groups of the form G = G1 x⋯x Gn, our construction has size poly(maxi |G i|, n, ε-1), and we show that a specific set S ⊂ Gn considered by Meka and Zuckerman that fools read-once branching programs over G is also ε-biased in this sense. For solvable groups whose abelian quotients have constant exponent, we obtain ε-biased sets of size (log|G|)1+o(1) poly(ε-1). Our techniques include derandomized squaring (in both the matrix product and tensor product senses) and a Chernoff-like bound on the expected norm of the product of independently random operators that may be of independent interest. © 2013 Springer-Verlag.