Design of MDP Convolutional Codes and Maximally Recoverable Codes Through the Lens of Matrix Completion

Abstract: 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.