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 $1 - H_q(p) - \epsilon$ are $(p, O(q H_q(p)/\epsilon))$-list decodable with high probability for all values of $p, q, \epsilon$, including the high error regime. This nearly matches the list size lower bound of $H_q(p)/\epsilon$ 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.