For a growing number of applications such as cellular, peer-to-peer, andsensor networks, efficient error-free transmission of data through a network isessential. Toward this end, K\"{o}tter and Kschischang propose the use ofsubspace codes to provide error correction in the network coding context. Theprimary construction for subspace codes is the lifting of rank-metric or matrixcodes, a process that preserves the structural and distance properties of theunderlying code. Thus, to characterize the structure and error-correctingcapability of these subspace codes, it is valuable to perform such acharacterization of the underlying rank-metric and matrix codes. This paperlays a foundation for this analysis through a framework for classifyingrank-metric and matrix codes based on their structure and distance properties. To enable this classification, we extend work by Berger on equivalence forrank-metric codes to define a notion of equivalence for matrix codes, and wecharacterize the group structure of the collection of maps that preserve suchequivalence. We then compare the notions of equivalence for these two relatedtypes of codes and show that matrix equivalence is strictly more general thanrank-metric equivalence. Finally, we characterize the set of equivalence mapsthat fix the prominent class of rank-metric codes known as Gabidulin codes. Inparticular, we give a complete characterization of the rank-metric automorphismgroup of Gabidulin codes, correcting work by Berger, and give a partialcharacterization of the matrix-automorphism group of the expanded matrix codesthat arise from Gabidulin codes.
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