AG Codes Achieve List-Decoding Capacity Over Constant-Sized Fields

The recently-emerging field of higher order MDS codes has sought to unify a number of concepts in coding theory. Such areas captured by higher order MDS codes include maximally recoverable (MR) tensor codes, codes with optimal list-decoding guarantees, and codes with constrained generator matrices (as in the GM-MDS theorem). By proving these equivalences, Brakensiek et al. (2024) showed the existence of optimally list-decodable Reed-Solomon codes over exponential sized fields. Building on this, recent breakthroughs by Guo and Zhang (2023) and Alrabiah et al. (2024) have shown that randomly punctured Reed-Solomon codes achieve list-decoding capacity (which is a relaxation of optimal list-decodability) over linear size fields. We extend these works by developing a formal theory of relaxed higher order MDS codes. In particular, we show that there are two inequivalent relaxations which we call lower and upper relaxations. The lower relaxation is equivalent to relaxed optimal list-decodable codes and the upper relaxation is equivalent to relaxed MR tensor codes with a single parity check per column. We then generalize the techniques of Guo-Zhang and Alrabiah-Guruswami-Li to show that both these relaxations can be constructed by randomly puncturing suitable algebraic-geometric codes over constant size fields. For this, we crucially use the generalized GM-MDS theorem for polynomial codes recently proved by Brakensiek et al. (2024). We obtain the following corollaries from our main result: 1) randomly punctured algebraic-geometric codes of rate R are list-decodable up to radius $\frac {L}{L+1}(1-R-\epsilon)$ with list size L over fields of size $\exp (O(L/\epsilon))$ . In particular, they achieve list-decoding capacity with list size $O(1/\epsilon)$ and field size $\exp (O(1/\epsilon ^{2}))$ . Prior to this work, AG codes were not even known to achieve list-decoding capacity; and 2) by randomly puncturing algebraic-geometric codes, we can construct relaxed MR tensor codes with a single parity check per column over constant-sized fields, whereas (non-relaxed) MR tensor codes require exponential field size.