Higher order MDS codes are an interesting generalization of MDS codes recently introduced by Brakensiek, Gopi and Makam (IEEE Trans. Inf. Theory 2022). In later works, they were shown to be intimately connected to optimally list-decodable codes and maximally recoverable tensor codes. Therefore (explicit) constructions of higher order MDS codes over small fields is an important open problem. Higher order MDS codes are denoted by $\operatorname{MDS}(\ell)$ where $\ell$ denotes the order of generality, $\operatorname{MDS}(2)$ codes are equivalent to the usual MDS codes. The best prior lower bound on the field size of an $(n,k)$-$\operatorname{MDS}(\ell)$ codes is $\Omega_\ell(n^{\ell-1})$, whereas the best known (non-explicit) upper bound is $O_\ell(n^{k(\ell-1)})$ which is exponential in the dimension. In this work, we nearly close this exponential gap between upper and lower bounds. We show that an $(n,k)$-$\operatorname{MDS}(3)$ codes requires a field of size $\Omega_k(n^{k-1})$, which is close to the known upper bound. Using the connection between higher order MDS codes and optimally list-decodable codes, we show that even for a list size of 2, a code which meets the optimal list-decoding Singleton bound requires exponential field size; this resolves an open question from Shangguan and Tamo (STOC 2020). We also give explicit constructions of $(n,k)$-$\operatorname{MDS}(\ell)$ code over fields of size $n^{(\ell k)^{O(\ell k)}}$. The smallest non-trivial case where we still do not have optimal constructions is $(n,3)$-$\operatorname{MDS}(3)$. In this case, the known lower bound on the field size is $\Omega(n^2)$ and the best known upper bounds are $O(n^5)$ for a non-explicit construction and $O(n^{32})$ for an explicit construction. In this paper, we give an explicit construction over fields of size $O(n^3)$ which comes very close to being optimal.

翻译：更高命令 MDS 代码是Brakesensiek、 Gopi 和 Makam 最近引入的 mDS 代码的有趣常规化 (Orkensiek $, Gopi $美元 。 Inf. Theory 2022) 。 在后来的作品中,它们被显示与最优列表可辨别的代码和最大可解析的高级代码密切连接。 因此( 明确) 在小域上, 更高命令MDS 代码的构建是一个重要的开放问题 。 更高命令MDS 代码由 $\ operatorname{MDS} (hell) 来表示, 美元表示通用的顺序, $O 最小值 美元 。 最高命令中, 最已知的( 非解析) 上一个( 未知的) 元码是 $Oell (n\\ kk), 在 Vemodeal- drode 字段中, 也显示一个最小的大小。