Random Reed-Solomon Codes are List Recoverable with Optimal List Size (2404.00206v2)

Abstract: We prove that Reed-Solomon (RS) codes with random evaluation points are list recoverable up to capacity with optimal output list size, for any input list size. Namely, given an input list size $\ell$, a designated rate $R$, and any $\varepsilon > 0$, we show that a random RS code is list recoverable from $1-R-\varepsilon$ fraction of errors with output list size $L = O(\ell/\varepsilon)$, for field size $q=\exp(\ell,1/\varepsilon) \cdot n^2$. In particular, this shows that random RS codes are list recoverable beyond the "list recovery Johnson bound". Such a result was not even known for arbitrary random linear codes. Our technique follows and extends the recent line of work on list decoding of random RS codes, specifically the works of Brakensiek, Gopi, and Makam (STOC 2023), and of Guo and Zhang (FOCS 2023).