Explicit Almost-Optimal $\varepsilon$-Balanced Codes via Free Expander Walks
Jun-Ting Hsieh, Sidhanth Mohanty, Rachel Yun Zhang
TL;DR
The paper addresses the open problem of constructing explicit binary codes with relative distance near $1/2$ that match the Gilbert–Varshamov bound in rate. It introduces free expander walks, running steps on a sequence of expanders drawn from near-Ramanujan graphs to amplify bias while avoiding alignment with any single graph; the construction leverages a base code with small bias and an all-signings near-Ramanujan ensemble to achieve an explicit code with rate $Ω(ε^{2+o(1)})$ and bias $≤ ε^{1-o(1)}$. This yields a simpler, explicit path to nearly optimal $\,ε$-balanced codes compared to Ta-Shma’s 2017 approach, with potential implications for decoding efficiency and pseudorandomness applications. The method combines an expander-walk lift with a carefully designed schedule $W^*$ to obtain strong operator-norm bounds on products of adjoined matrices, thereby establishing the desired distance–rate tradeoff in the low-rate regime.
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 $Ω(\varepsilon^{2+o(1)})$, 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 an arguably 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.