A code of length $n$ is said to be (combinatorially) $(\rho,L)$-list decodable if the Hamming ball of radius $\rho n$ around any vector in the ambient space does not contain more than $L$ codewords. We study a recently introduced class of higher order MDS codes, which are closely related (via duality) to codes that achieve a generalized Singleton bound for list decodability. For some $\ell\geq 1$, higher order MDS codes of length $n$, dimension $k$, and order $\ell$ are denoted as $(n,k)$-MDS($\ell$) codes. We present a number of results on the structure of these codes, identifying the `extend-ability' of their parameters in various scenarios. Specifically, for some parameter regimes, we identify conditions under which $(n_1,k_1)$-MDS($\ell_1$) codes can be obtained from $(n_2,k_2)$-MDS($\ell_2$) codes, via various techniques. We believe that these results will aid in efficient constructions of higher order MDS codes. We also obtain a new field size upper bound for the existence of such codes, which arguably improves over the best known existing bound, in some parameter regimes.

翻译：称长度为$n$的码为（组合意义上的）$(\rho,L)$-列表可译码，若任意向量在汉明球半径$\rho n$内包含的码字不超过$L$个。本文研究近期引入的一类高阶MDS码，该码类通过对偶性与达到列表可译广义Singleton界的码密切相关。对某些$\ell\geq 1$，称长度为$n$、维数为$k$、阶数为$\ell$的高阶MDS码为$(n,k)$-MDS($\ell$)码。我们从结构角度给出若干结果，刻画了其参数在不同场景下的"可扩展性"。具体而言，针对某些参数区间，我们确定了通过多种技术从$(n_2,k_2)$-MDS($\ell_2$)码构造$(n_1,k_1)$-MDS($\ell_1$)码的条件。我们相信这些结果将有助于高阶MDS码的高效构造。此外，我们获得了这类码存在性的新域大小上界，在某些参数区间内该上界优于现有最佳结果。