Bounds for List-Decoding and List-Recovery of Random Linear Codes

A family of error-correcting codes is list-decodable from error fraction p if, for every code in the family, the number of codewords in any Hamming ball of fractional radius p is less than some integer L. It is said to be list-recoverable for input list size \ell if for every sufficiently large subset of at least L codewords, there is a coordinate where the codewords take more than \ell values. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below q is the alphabet size, and \varepsilon > 0 is the gap to capacity). (1) A random linear code of rate 1 - \log {q}(\ell) - \varepsilon requires list size L \ge \ell {\Omega (1/ \varepsilon)} for list-recovery from input list size \ell . (2) A random linear code of rate 1 - h{q}(p) - \varepsilon requires list size L \ge \left \lfloor{ {h{q}(p)/ \varepsilon +0.99}}\right \rfloor for list-decoding from error fraction p. (3) A random binary linear code of rate 1 - h{2}(p) - \varepsilon is list-decodable from average error fraction p with list size with L \leq \left \lfloor{ {h{2}(p)/ \varepsilon }}\right \rfloor + 2. Our lower bounds follow by exhibiting an explicit subset of codewords so that this subset - or some symbol-wise permutation of it - lies in a random linear code with high probability. Our upper bound follows by strengthening a result of (Li, Wootters, 2018).