Products of MDS codes are of major practical importance; for a recent example, they are used in Data Availability Sampling (DAS) in blockchain networks such as Celestia and as part of the Ethereum roadmap. This motivates us to consider subcodes of such codes with the goal of obtaining a larger minimum distance. In this paper, we present explicit constructions of subcodes of Reed--Solomon product codes, along with bounds on their minimum distance. In particular, they achieve an optimal or near-optimal dimension--distance tradeoff. For component codes of dimension $r$, our construction requires a field whose size is bounded linearly by the overall product code length, and attains the maximum possible minimum distance for subcode dimensions $r^2-1$, $r^2-2$, and all dimensions at most $2r-1$. Furthermore, we establish a new upper bound on the minimum distance of subcodes of the product of two codes with identical parameters.

翻译：MDS码的乘积具有重要的实际意义。近期的一个例子是，它们被用于Celestia等区块链网络中的数据可用性采样(DAS)，并作为以太坊路线图的一部分。这促使我们考虑这类码的子码，以获取更大的最小距离。本文给出了里德-所罗门乘积码子码的显式构造，以及其最小距离的界。特别地，这些构造实现了最优或接近最优的维数-距离权衡。对于维数为$r$的分量码，我们的构造所需的域大小与整体乘积码长度呈线性关系，并在子码维数为$r^2-1$、$r^2-2$以及所有不超过$2r-1$的维数下达到最大可能的最小距离。此外，我们建立了两个具有相同参数码的乘积子码的最小距离的新上界。