The list size of random linear codes at capacity
Let $C \le \mathbb{F}_q^n$ be a uniformly random $\mathbb{F}_q$-linear code of rate $1 - h_q(ρ) - \varepsilon$, and let $L^*(C,ρ)$ be the least $L$ such that every Hamming ball of relative radius $ρ$ contains at most $L$ codewords of $C$. That $L^* = Θ_{q,ρ}(1/\varepsilon)$ has been known since work of Guruswami, Håstad and Kopparty and of Guruswami and Narayanan. Guruswami, Li, Mosheiff, Resch, Silas and Wootters proved that the constant in front of $1/\varepsilon$ is at least $h_q(ρ)$ for all $q$, along with an upper bound special to $q = 2$ which narrowed $L^*$ to within three consecutive integers in that case. But for $q \ge 3$ no upper bound with the correct constant was known. We determine $L^*$ for every prime power $q$. Let $ζ:= h_q(ρ)/\varepsilon$. For every sufficiently small $\varepsilon$, with probability $1-o(1)$ over the choice of $C$, $$L^*(C,ρ) = \lceil ζ\rceil,$$ unless the fractional part of $ζ$ is at most $q^{-Ω_{q,ρ}(ζ)}$, in which case $L^*(C,ρ)$ is $\lfloor ζ\rfloor$ or $\lfloor ζ\rfloor + 1$. By the threshold characterization of random linear codes due to Mosheiff, Resch, Ron-Zewi, Silas and Wootters, both bounds reduce to a two-sided estimate of a single quantity $V(q,L,ρ)$, where $1-V(q,L,ρ)$ is the threshold rate for $(ρ,L)$-list-decodability. We prove for all large $L$: $$h_q(ρ)(1 + 1/L) - q^{-Ω_{q,ρ}(L)} \le V(q,L,ρ) \le h_q(ρ)(1 + 1/L).$$ The upper bound rests on a new entropy inequality for sparse random vectors under pairwise non-proportional linear constraints, proved with the Erdős-Rado sunflower lemma. The lower bound is an exact analysis of the distribution introduced by Guruswami, Li, Mosheiff, Resch, Silas and Wootters.