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 GZ and AGL 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. First, randomly punctured AG codes of rate $R$ 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. Second, by randomly puncturing AG 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.

翻译：新兴的高阶MDS码领域致力于统一编码理论中的多个概念。这些由高阶MDS码涵盖的领域包括最大可恢复（MR）张量码、具有最优列表译码保证的码，以及具有约束生成矩阵的码（如GM-MDS定理所述）。Brakensiek-Gopi-Makam通过证明这些等价性，展示了在指数规模域上存在最优列表可译码的里德-所罗门码。在此基础上，Guo-Zhang和Alrabiah-Guruswami-Li的最新突破表明，在线性规模域上，随机穿刺的里德-所罗门码能达到列表译码容量（这是最优列表可译性的松弛形式）。我们通过发展一套严格的松弛高阶MDS码理论来扩展这些工作。特别地，我们证明存在两种不等价的松弛形式，分别称为下松弛和上松弛。下松弛等价于松弛最优列表可译码，而上松弛等价于每列具有单个奇偶校验的松弛MR张量码。随后，我们推广了GZ和AGL的技术，表明通过随机穿刺合适的代数几何码，这两种松弛形式均可在常数规模域上构造。为此，我们关键性地利用了Brakensiek-Dhar-Gopi最近证明的针对多项式码的广义GM-MDS定理。从我们的主要结果中，得到以下推论。首先，速率为$R$的随机穿刺AG码在列表大小为$O(1/\epsilon)$、域大小为$\exp(O(1/\epsilon^2))$时达到列表译码容量。在本工作之前，AG码甚至未被证明能达到列表译码容量。其次，通过随机穿刺AG码，我们可以在常数规模域上构造每列具有单个奇偶校验的松弛MR张量码，而（非松弛）MR张量码需要指数规模的域。