The matrix completion problem provides a unifying lens through which many fundamental problems in coding theory can be viewed. In this paper, we investigate Locally Recoverable Codes (LRCs) with Maximal Recoverability (MR) and Maximum Distance Profile (MDP) convolutional codes in the framework of matrix completion. In particular, we present techniques that are general enough to provide constructions for both types of codes. A common feature of our code constructions is the sparsity of their generator matrices and the property that a large number of the entries of the generator matrices are elements of a small subfield of a larger extension field.

翻译：矩阵补全问题为编码理论中的许多基本问题提供了统一的视角。本文在矩阵补全框架下研究具有最大可恢复性的本地可恢复码以及最大距离轮廓卷积码。具体而言，我们提出了足够通用的技术，能够同时为这两类码提供构造方法。我们构造的码具有一个共同特征：其生成矩阵是稀疏的，且生成矩阵中的大量元素属于较小子域，而非较大的扩域。