Universal combinatorial list-decodability for finite simple groups

Let $G$ be a finite simple group, let $H$ be an arbitrary finite or infinite group, and let $\operatorname{aHom}(G,H)$ denote the homomorphism code. A class of finite groups is universally CombEcon if, for every group in the class and every such codomain $H$, the corresponding homomorphism code is CombEcon; equivalently, for every $\varepsilon>0$, every received word has at most $\operatorname{poly}(1/\varepsilon)$ codewords within distance $\textsf{mindist}-\varepsilon$, where $\textsf{mindist}$ is the code's minimum distance.

Universal finite-simple-group conjecture. The class of finite simple groups is universally CombEcon.

The paper proves this property for finite abelian and alternating groups, identifying the alternating groups as a first test case for the expected general phenomenon among finite simple groups. The conjecture extends the combinatorial list-decoding result to all finite simple domains, while allowing arbitrary codomains.