The random linear-code list-decoding conjecture

Let $q$ be a prime power, let $0<p<1-1/q$, and let $H_q(p)$ denote the $q$-ary entropy. A random linear code of rate $1-H_q(p)-\delta$ is chosen uniformly among linear codes of that rate over $\bF_q^n$, and a code is $(p,L)$-list decodable if every Hamming ball of relative radius $p$ contains at most $L$ codewords. Random linear-code list-decoding conjecture. For every $\delta>0$, a random linear code of rate $1-H_q(p)-\delta$ over $\bF_q$ is $(p,1/\delta)$-list decodable with high probability as $n\to\infty$. Random linear codes are a natural structured analogue of uniformly random codes; the conjecture asserts constant-in-$n$ list size with the expected dependence on the gap from capacity.