Asymptotically Optimal List Size of Random Linear Codes

Abstract

We prove that for every fixed prime power $q$, every $p\in(0,1-1/q)$, and every $\varepsilon>0$ with $1-H_q(p)-\varepsilon>0$, a random linear code over $\mathbb{F}_q$ of rate $1-H_q(p)-\varepsilon$ is $(p,\,\left\lceil\frac{H_q(p)}{\varepsilon}\right\rceil+O_{p,q}(1))\text{-list-decodable}$ with probability at least $1-q^{-Ω(n)}$. Guruswami, Li, Mosheiff, Resch, Silas, and Wootters showed that, for sufficiently small $\varepsilon$, random linear codes require list size at least $\left\lfloor\frac{H_q(p)}{\varepsilon}+0.99\right\rfloor,$ and conjectured that $\frac{H_q(p)}{\varepsilon}(1+o(1))$ suffices as $\varepsilon\to 0$. This conjecture was previously known for $q=2$, where the upper bound $H_2(p)/\varepsilon+2$ was established. For $q>2$, however, the best known upper bound was $C_{p,q}/\varepsilon$ for a constant $C_{p,q}$ depending on $p$ and $q$. Our result resolves the conjecture for every prime power $q$ and, in fact, establishes the sharper upper bound $\frac{H_q(p)}{\varepsilon}+O_{p,q}(1)$.