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, number codewords any Hamming ball fractional radius less than some integer $L$ that independent length. It said to be list-recoverable input list size $\ell$ if sufficiently large subset (of or more), there a coordinate where take more values. The parameter either case. capacity, i.e., largest possible rate these notions as $L \to \infty$, known $1-h_q(p)$ list-decoding, and $1-\log_q \ell$ list-recovery, $q$ alphabet family. In this work, we study random linear both list-decoding list-recovery approaches capacity. We show following claims hold with high probability over choice (below, $\epsilon > 0$ gap capacity). (1) $1 - \log_q(\ell) \epsilon$ requires \ge \ell^{\Omega(1/\epsilon)}$ $\ell$. This surprisingly contrast completely codes, = O(\ell/\epsilon)$ suffices w.h.p. (2) h_q(p) \lfloor h_q(p)/\epsilon+0.99 \rfloor$ $p$, when $\epsilon$ small. (3) binary h_2(p) average \leq h_2(p)/\epsilon \rfloor + 2$. The second third results together precisely pin down sizes average-radius three