Abstract

The rank metric measures the distance between two matrices by the rank of their difference. Codes designed for the rank metric have attracted considerable attention in recent years, reinforced by network coding and further motivated by a variety of applications. In code-based cryptography, the hardness of the corresponding generic decoding problem can lead to systems with reduced public-key size. In distributed data storage, codes in the rank metric have been used repeatedly to construct codes with locality, and in coded caching, they have been employed for the placement of coded symbols. This survey gives a general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research.

Rank-Metric Codes and Their Applications

Rank-metric codes date back to the 1970s and today play a vital role in many areas of coding theory and cryptography. In this survey the authors provide a comprehensive overview of the known properties of rank-metric codes and their applications.

The authors begin with an accessible and complete introduction to rank-metric codes, their properties and their decoding. They then discuss at length rank-metric code-based quantum resistant encryption and authentication schemes. The application of rank-metric codes to distributed data storage is also outlined. Finally, the constructions of network codes based on MRD codes, constructions of subspace codes by lifting rank-metric codes, bounds on the cardinality, and the list decoding capability of subspace codes is covered in depth.

Rank-Metric Codes and Their Applications provides the reader with a concise, yet complete, general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research.