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 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.

翻译：我们研究在低码率、高距离条件下构造显式码的问题，要求其码率和距离匹配Gilbert-Varshamov界。2017年，Ta-Shma给出了一族显式码：码字间相对距离为$\frac{1-\varepsilon}{2}$，码率可达$Ω(\varepsilon^{2+o(1)})$，在$\varepsilon^{o(1)}$因子内匹配Gilbert-Varshamov界。Ta-Shma的构造基于优质码出发，利用$s$-宽替换乘积产生的游走放大偏差。本文给出一种更简单的近优构造，其核心是称为“自由扩展图游走”的方法：普通扩展图游走中，每一步选取精心设计序列中的不同扩展图。该扩展图序列源自O'Donnell和Wu提出的近$X$-Ramanujan图构造。此外，我们还将讨论近$X$-Ramanujan图在“平均意义”无损扩张与旋转扩展图方面的其他应用。