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.
Combinatorial list-decoding of Reed-Solomon codes beyond the Johnson radius
自Guruswami和Sudan的工作以来，Johnson半径以外的RS码的列表译码一直是一个主要的开放问题。Rudra和Wootters使用高维概率的技术，在足够大的字母表上，大多数RS码确实可以在这个半径之外进行列表解码。在本文中，我们采取了一个更组合的方法，使我们能够确定精确的关系（精确常数）之间的解码半径和列表大小。我们证明了一个广义的Singleton界给定的列表大小，并猜想界是紧的大多数RS码在足够大的有限域。我们还表明，猜想成立的列表大小$2 \text{ and }3$，并作为一个产品表明，大多数RS码的速率至少$1/9$是列表解码超出约翰逊半径。最后，我们给出了这种RS码的第一个显式构造。在证明中使用的主要工具是一种新的类型的线性依赖代码之间的代码，包含在一个小的汉明球，和循环空间的概念，从图论。这两个都没有在列表解码的上下文中使用过。
Stein Faculty of Computer and Information Science, Ben-Gurion University（本-古里安大学斯坦计算机与信息科学学院）；Instituto de Telecomunicações and Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院电信研究所与数学系）；Informatics Institute, University of Amsterdam（阿姆斯特丹大学信息学研究所）
