We introduce a novel family of expander-based error correcting codes. These codes can be sampled with randomness linear in the block-length, and achieve list-decoding capacity (among other local properties). Our expander-based codes can be made starting from any family of sufficiently low-bias codes, and as a consequence, we give the first construction of a family of algebraic codes that can be sampled with linear randomness and achieve list-decoding capacity. We achieve this by introducing the notion of a pseudorandom puncturing of a code, where we select $n$ indices of a base code $C\subset \mathbb{F}_q^m$ via an expander random walk on a graph on $[m]$. Concretely, whereas a random linear code (i.e. a truly random puncturing of the Hadamard code) requires $O(n^2)$ random bits to sample, we sample a pseudorandom linear code with $O(n)$ random bits. We show that pseudorandom puncturings satisfy several desirable properties exhibited by truly random puncturings. In particular, we extend a result of (Guruswami Mosheiff FOCS 2022) and show that a pseudorandom puncturing of a small-bias code satisfies the same local properties as a random linear code with high probability. As a further application of our techniques, we also show that pseudorandom puncturings of Reed Solomon codes are list-recoverable beyond the Johnson bound, extending a result of (Lund Potukuchi RANDOM 2020). We do this by instead analyzing properties of codes with large distance, and show that pseudorandom puncturings still work well in this regime.

翻译：我们引入了一类基于扩展器的纠错码新族。这类码可用与码长呈线性关系的随机性进行采样，并能在实现列表解码容量的同时满足其他局部性质。通过从任意具有足够低偏置的码族出发构造基于扩展器的码，我们首次实现了可用线性随机性采样且达到列表解码容量的代数码族构造。为此我们提出"伪随机打孔"概念——通过扩展随机游走选取基码 $C\subset \mathbb{F}_q^m$ 在 $[m]$ 上的 $n$ 个索引。具体而言，当随机线性码（即哈达玛码的真正随机打孔）需要 $O(n^2)$ 随机比特进行采样时，我们仅用 $O(n)$ 随机比特即可采样伪随机线性码。研究表明，伪随机打孔满足真正随机打孔的多个理想性质。特别地，我们推广了(Guruswami Mosheiff FOCS 2022)的结论：低偏置码的伪随机打孔能以高概率保持与随机线性码相同的局部性质。作为技术的进一步应用，我们还证明里德-所罗门码的伪随机打孔可在约翰逊界之外实现列表可恢复性，推广了(Lund Potukuchi RANDOM 2020)的结论。这通过分析具有大距离码的性质实现，并证明伪随机打孔在此场景下仍能有效运作。