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 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 epsilon > 0 is the gap to capacity). (1) A random linear code of rate 1 - log(q) (l) - epsilon requires list size L >= l(Omega(1/epsilon)) for list-recovery from input list size l. (2) A random linear code of rate 1 - h(q)(p) - epsilon requires list size L >= left perpendicularh(q)(p)/epsilon + 0.99right perpendicular for list-decoding from error fraction p. (3) A random binary linear code of rate 1-h(2)(p)-epsilon is list-decodable from average error fraction p with list size with L <= left perpendicularh(2)(p)/epsilon right perpendicular + 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).

Automated summary

This work investigates the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. Lower and upper bounds are obtained by exhibiting explicit subsets of codewords and strengthening existing results.