Analyzing Ta-Shma's Code via the Expander Mixing Lemma

TL;DR

This paper offers a combinatorial perspective on Ta-Shma's code construction using the expander mixing lemma, providing new insights and an alternative proof of the expander hitting set lemma.

Contribution

It rederives Ta-Shma's analysis from a combinatorial viewpoint and introduces an alternative proof of the expander hitting set lemma.

Findings

Rederived Ta-Shma's analysis combinatorially

Provided an alternative proof of the expander hitting set lemma

Enhanced understanding of expander-based code constructions

Abstract

Random walks in expander graphs and their various derandomizations (e.g., replacement/zigzag product) are invaluable tools from pseudorandomness. Recently, Ta-Shma used s-wide replacement walks in his breakthrough construction of a binary linear code almost matching the Gilbert-Varshamov bound (STOC 2017). Ta-Shma's original analysis was entirely linear algebraic, and subsequent developments have inherited this viewpoint. In this work, we rederive Ta-Shma's analysis from a combinatorial point of view using repeated application of the expander mixing lemma. We hope that this alternate perspective will yield a better understanding of Ta-Shma's construction. As an additional application of our techniques, we give an alternate proof of the expander hitting set lemma.