Codes in the Sum-Rank Metric: Fundamentals and Applications

Codes in the sum-rank metric have attracted significant attention for their applications in distributed storage systems, multishot network coding, streaming over erasure channels, and multi-antenna wireless communication. This monograph provides a tutorial introduction to the theory and applications of sum-rank metric codes over finite fields. At the heart of the monograph is the construction of linearized Reed–Solomon codes, a general construction of maximum sum-rank distance (MSRD) codes with polynomial field sizes. Linearized Reed–Solomon codes specialize to classical Reed–Solomon and Gabidulin code constructions in the Hamming and rank metrics, respectively, and they admit an efficient Welch–Berlekamp decoding algorithm. Applications of these codes in distributed storage systems, network coding, and multi-antenna communication are developed. Other families of codes in the sum-rank metric, including convolutional codes and subfield subcodes are described, and recent results in the general theory of codes in the sum-rank metric are surveyed.