Explicit Almost-Optimal $\varepsilon$-Balanced Codes via Free Expander Walks

TL;DR

This paper introduces a simpler explicit construction of near-optimal error-correcting codes using free expander walks, matching the Gilbert-Varshamov bound in the low-rate, high-distance regime.

Contribution

It presents a novel, simplified method for constructing explicit codes with optimal parameters based on free expander walks and near-$X$-Ramanujan graphs.

Findings

Constructed codes match the Gilbert-Varshamov bound up to small factors.

Introduced the concept of free expander walks for code construction.

Discussed applications of near-$X$-Ramanujan graphs to expanders.

Abstract

We study the problem of constructing explicit codes whose rate and distance match the Gilbert-Varshamov bound in the low-rate, high-distance regime. In 2017, Ta-Shma gave an explicit family of codes where every pair of codewords has relative distance $\frac{1-\varepsilon}{2}$, with rate $\Omega(\varepsilon^{2+o(1)})$, matching the Gilbert-Varshamov bound up to a factor of $\varepsilon^{o(1)}$. Ta-Shma's construction was based on starting with a good code and amplifying its bias with walks arising from the $s$-wide-replacement product. In this work, we give a simpler almost-optimal construction, based on what we call free expander walks: ordinary expander walks where each step is taken on a distinct expander from a carefully chosen sequence. This sequence of expanders is derived from the construction of near-$X$-Ramanujan graphs due to O'Donnell and Wu. We additionally discuss some additional applications of near-$X$-Ramanujan graphs to "on average" lossless expansion and rotating expanders.