List Decoding, Linear Hashing, and Furstenberg over $\mathbb{F}_q$

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 probabilit