Near-Optimal Expanding Generating Sets for Solvable Permutation Groups

TL;DR

This paper presents a deterministic polynomial-time algorithm for constructing small, expanding generating sets for solvable permutation groups, leading to new explicit constructions of epsilon-bias spaces with improved size bounds.

Contribution

It introduces a novel polynomial-time method to find expanding generating sets of size nearly quadratic for solvable permutation groups, and derives improved epsilon-bias space constructions.

Findings

Constructed expanding generating sets of size n^2 polylogarithmic factors for solvable groups.

Provided explicit epsilon-bias space constructions with size n polylogarithmic factors.

Achieved spectral expansion properties in Cayley graphs of solvable groups.

Abstract

Let $G =<S>$ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n^2)$ for $G$. More precisely, the algorithm computes a subset $T\subset G$ of size $\tilde{O}(n^2)(1/\lambda)^{O(1)}$ such that the undirected Cayley graph $Cay(G,T)$ is a $\lambda$-spectral expander (the $\tilde{O}$ notation suppresses $\log ^{O(1)}n$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n\poly(\log d))(\frac{1}{\varepsilon})^{O(1)}$ for the groups $\Z_d^n$. The earlier known size bound was $O((d+n/\varepsilon^2))^{11/2}$ given by \cite{AMN98}.