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

2026-04-23

AI Summary

This work proposes a unified construction framework based on matrix completion that simultaneously addresses the design of maximum distance profile (MDP) convolutional codes and maximally recoverable (MR) locally recoverable codes (LRCs). By introducing a sparse generator matrix structure and extensively employing elements from a small subfield within a large extension field, the approach significantly reduces construction complexity while preserving optimal distance and recovery properties. For the first time, matrix completion is leveraged as a common perspective to unify these two distinct code families. Integrating algebraic coding theory, finite field extensions, and sparse matrix design, this study enables efficient and unified constructions of two important classes of error-correcting codes, offering a novel paradigm for reliable storage and communication systems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Applications of cryptography

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.

Problem

• Matrix Completion
• Locally Recoverable Codes
• Maximal Recoverability
• MDP Convolutional Codes
• Coding Theory

Innovation

• Matrix Completion
• Maximally Recoverable Codes
• MDP Convolutional Codes
• Sparse Generator Matrices
• Subfield Elements