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：摘要：我们引入了一类基于扩展器的全新纠错码族。这类码能以块长线性随机性进行采样，并达到列表译码容量（兼具其他局部性质）。我们的基于扩展器的纠错码可从任意低偏置码族出发构造，由此首次给出能以线性随机性采样并实现列表译码容量的一类代数码族。实现这一目标的关键在于引入码的伪随机打孔概念——通过图 $[m]$ 上的扩展器随机游走选取基码 $C\subset \mathbb{F}_q^m$ 的 $n$ 个索引。具体而言，当随机线性码（即阿达马码的真随机打孔）需要 $O(n^2)$ 随机比特进行采样时，我们通过 $O(n)$ 随机比特即可采样伪随机线性码。研究表明，伪随机打孔满足真随机打孔展现的若干理想性质。特别地，我们推广了(Guruswami Mosheiff FOCS 2022)的结论，证明小偏置码的伪随机打孔能以高概率满足与随机线性码相同的局部性质。作为该技术的进一步应用，我们还证明里德-所罗门码的伪随机打孔可实现超越约翰逊界的列表恢复性，从而推广了(Lund Potukuchi RANDOM 2020)的结论。这一推广通过分析大距离码的性质完成，并证明伪随机打孔在该场景下仍然有效。