AI 中文总结
该研究解决了随机线性码列表译码的渐近最优列表大小猜想，对任意素数幂q，将其上界精确为H_q(p)/ε+O_{p,q}(1)，改进了q>2时的此前结果。
AI 中文摘要
我们证明，对于每个固定素数幂q、每个p∈(0,1−1/q)以及每个满足1−H_q(p)−ε>0的ε>0，速率为1−H_q(p)−ε的有限域F_q上的随机线性码，以至少q^{−Ω(n)}的概率是(p, ⌈H_q(p)/ε⌉+O_{p,q}(1))-列表可译码的。Guruswami、Li、Mosheiff、Resch、Silas和Wootters指出，对于足够小的ε，随机线性码需要至少⌊H_q(p)/ε+0.99⌋的列表大小，并猜想当ε→0时，H_q(p)/ε(1+o(1))就足够了。该猜想此前仅在q=2时成立，此时已确立上界H_2(p)/ε+2；而对于q>2，已知的最佳上界为C_{p,q}/ε，其中C_{p,q}是依赖于p和q的常数。我们的结果解决了对每个素数幂q的该猜想，实际上确立了更精确的上界H_q(p)/ε+O_{p,q}(1)。
英文摘要
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 $\frac{H_q(p)}{\varepsilon}+O_{p,q}(1)$.
Comments12 pages
