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

Random walks in expander graphs and their various derandomizations (e.g., replacement/zig-zag 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.