发表机构
Simons Institute for the Theory of Computing, UC Berkeley; University of Texas at Austin（加州大学伯克利分校西蒙斯理论计算研究所; 德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出重数间隙多项式方法，证明了有限域上随机线性码列表可解码性和线性哈希最大负载的新上界，几乎匹配已知下界，并统一改进了此前结果。
AI 中文摘要
我们给出了在任意有限域 $\mathbb{F}_q$ 上，容量处随机线性码的列表大小、线性哈希函数的最大负载以及 Furstenberg 集合的新界。1. 在 $\mathbb{F}_q$ 上，码率为 $1 - H_q(p) - \epsilon$ 的随机线性码，对于所有 $p, q, \epsilon$ 的取值（包括高错误率区间），以高概率是 $(p, O(q H_q(p)/\epsilon))$-列表可解码的。这几乎匹配了 Guruswami, Li, Mosheiff, Resch, Silas 和 Wootters [IEEE Trans. Inf. Theory 2022] 给出的列表大小下界 $H_q(p)/\epsilon$。我们的界是自 Guruswami, Håstad 和 Kopparty [STOC 2010] 以来，对 $q > 2$ 的首次统一改进。2. 在 $\mathbb{F}_q$ 上，将 $n$ 个球哈希到 $n$ 个箱子的线性哈希函数，在期望意义下以及以 $1-o(1)$ 的概率，都能达到最大负载 $O(q \ln \ln q / {\ln q}) \cdot \ln n / {\ln \ln n}$。这几乎匹配了下界 $\ln n / {\ln \ln n}$。此前，对于 $q > 2$，仅有 Alon, Dietzfelbinger, Miltersen, Petrank 和 Tardos [J. ACM 1999] 给出的多对数上界。我们将列表可解码性和线性哈希归结为强 Furstenberg 集合下界，并利用一种新的重数间隙多项式方法证明了该下界。以往的多项式方法通过研究在集合 $S$ 上消失的多项式来分析 $S$，而我们考虑的是处处消失但在 $S$ 内部比外部具有更高重数的多项式。
英文摘要
We give new bounds for list sizes of random linear codes at capacity, max loads of linear hash functions, and Furstenberg sets, over every finite field $\mathbb{F}_q$. 1. Random linear codes over $\mathbb{F}_q$ with rate $1 - H_q(p) - ε$ are $(p, O(q H_q(p)/ε))$-list decodable with high probability for all values of $p, q, ε$, including the high error regime. This nearly matches the list size lower bound of $H_q(p)/ε$ due to Guruswami, Li, Mosheiff, Resch, Silas, and Wootters [IEEE Trans. Inf. Theory 2022]. Our bound is the first uniform improvement for $q > 2$ since Guruswami, Håstad, and Kopparty [STOC 2010]. 2. Linear hash functions over $\mathbb{F}_q$ hashing $n$ balls to $n$ bins achieve maximum load $O(q \ln \ln q / {\ln q}) \cdot \ln n / {\ln \ln n}$, both in expectation and with probability $1-o(1)$. This nearly matches the lower bound of $\ln n / {\ln \ln n}$. Previously, only a polylogarithmic upper bound was known for $q > 2$, due to Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos [J. ACM 1999]. We reduce list decodability and linear hashing to strong Furstenberg set lower bounds, which we prove using a new polynomial method of multiplicity gaps. While previous polynomial methods analyze a set $S$ by studying polynomials that vanish on it, we consider polynomials that vanish everywhere, but with higher multiplicity inside $S$ than outside.
