π€ AI Summary
This work resolves the long-standing problem of achieving list-decoding capacity with algebraic geometry (AG) codes over constant-size finite fields. Addressing the limitation that classical AG codes fail to attain capacity over small fields, we introduce a random puncturing technique and establish, for the first time, a bipartite theoretical framework for relaxed high-order MDS codesβdefining two inequivalent relaxations: one yielding optimally list-decodable codes, the other enabling single-check-column relaxed maximally recoverable (MR) tensor codes. We prove that randomly punctured AG codes achieve list decoding radius $L/(L+1)(1-R-varepsilon)$ at any rate $R$, with list size $O(1/varepsilon)$, over fields of size $exp(O(1/varepsilon^2))$. Furthermore, we construct relaxed MR tensor codes over constant-size fields, breaking the exponential field-size barrier inherent in traditional MR codes.
π Abstract
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-Gopi-Makam showed the existence of optimally list-decodable Reed-Solomon codes over exponential sized fields. Building on this, recent breakthroughs by Guo-Zhang and Alrabiah-Guruswami-Li 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 over constant size fields by randomly puncturing suitable algebraic-geometric codes. For this, we crucially use the generalized GM-MDS theorem for polynomial codes recently proved by Brakensiek-Dhar-Gopi. We obtain the following corollaries from our main result: Randomly punctured algebraic-geometric codes of rate R are list-decodable up to radius L/L+1(1βRβΡ) with list size L over fields of size exp(O(L/Ρ)). In particular, they achieve list-decoding capacity with list size O(1/Ρ) and field size exp(O(1/Ρ2)). Prior to this work, AG codes were not even known to achieve list-decoding capacity. 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.
