Bounds for list-decoding and list-recovery of random linear codes
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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 that is independent of the code length. It is said to be list-recoverable for input list size ℓ if for every sufficiently large subset of codewords (of size L or more), there is a coordinate where the codewords take more than ℓ values. The parameter L is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size L →∞, is known to be 1-h_q(p) for list-decoding, and 1-log_q ℓ for list-recovery, where q is the alphabet size of the code family. 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, ϵ > 0 is the gap to capacity). (1) A random linear code of rate 1 - log_q(ℓ) - ϵ requires list size L >ℓ^Ω(1/ϵ) for list-recovery from input list size ℓ. This is surprisingly in contrast to completely random codes, where L = O(ℓ/ϵ) suffices w.h.p. (2) A random linear code of rate 1 - h_q(p) - ϵ requires list size L > h_q(p)/ϵ+0.99 for list-decoding from error fraction p, when ϵ is sufficiently small. (3) A random binary linear code of rate 1 - h_2(p) - ϵ is list-decodable from average error fraction p with list size with L ≤ h_2(p)/ϵ + 2. The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.
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