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Description:
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 ⊂ 𝔽_q^m in a correlated fashion. Concretely, whereas a random linear code (i.e. a truly random puncturing of the Hadamard code) requires O(n log(m)) random bits to sample, we sample a pseudorandom linear code with O(n + log (m)) random bits by instantiating our pseudorandom puncturing as a length n random walk on an exapnder graph on [m]. In particular, we extend a result of Guruswami and 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 and 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.