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.

翻译：本研究探讨在低码率、高距离区间内构造码率与距离均达到吉尔伯特-瓦尔沙莫夫界的显式编码问题。2017年，Ta-Shma提出了一类显式编码族，其中任意两个码字间的相对距离为$\frac{1-\varepsilon}{2}$，码率达到$Ω(\varepsilon^{2+o(1)})$，该结果与吉尔伯特-瓦尔沙莫夫界仅相差$\varepsilon^{o(1)}$因子。Ta-Shma的构造方法基于初始优质码，并利用源于$s$宽置换积的游走过程进行偏置放大。本工作提出了一种可论证更简洁的近似最优构造方案，其核心是我们提出的自由扩展图游走：在精心选取的扩展图序列中，每个游走步骤都在不同的扩展图上进行。该扩展图序列源自O'Donnell与Wu提出的近似$X$-拉马努金图构造方法。