Bounds for list-decoding and list-recovery of random linear codes

A family of error-correcting codes is listdecodable 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 ℓ if for every sufficiently large subset of at least L codewords, there is a coordinate where the codewords take more than ℓ 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 ε > 0 is the gap to capacity). (1) A random linear code of rate 1 - logq(ℓ)-ε requires list size L ≥ ℓΩ(1/ε) for list-recovery from input list size ℓ. (2) A random linear code of rate 1 - hq(p) - ε requires list size L ≥ ⌊hq(p)/ε + 0.99⌋ for list-decoding from error fraction p. (3) A random binary linear code of rate 1 - h2(p) - ε is list-decodable from average error fraction p with list size with L ≤ ⌊h2(p)/ε⌋ + 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).