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.
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