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Equivalence and duality for rank-metric and matrix codes
For a growing number of applications such as cellular, peer-to-peer, and sensor networks, efficient error-free transmission of data through a network is essential. Toward this end, Kötter and Kschischang propose the use of subspace codes to provide error correction in the network coding context. The primary construction for subspace codes is the lifting of rank-metric or matrix codes, a process that preserves the structural and distance properties of the underlying code. Thus, to characterize the structure and error-correcting capability of these subspace codes, it is valuable to perform such a characterization of the underlying rank-metric and matrix codes. This dissertation lays a foundation for this analysis through a framework for classifying rank-metric and matrix codes based on their structure and distance properties, and performs such a classification for a particular subset of matrix codes. To enable this classification, we extend work by Berger on equivalence for rank-metric codes to define a notion of equivalence for matrix codes, and we characterize the group structure of the collection of maps that preserve such equivalence. Following Berger, we investigate the subset of equivalence maps that fix an important class of rank-metric codes known as Gabidulin codes. We correct Berger's characterization of the linear rank-metric automorphism group of Gabidulin codes, and extend this analysis to the case of matrix codes that arise by expanding Gabidulin codes. With these notions of equivalence in place, we turn to performing a complete classification of the equivalence classes of self-dual matrix codes. We focus on self-dual matrix codes in part because a classification of self-dual codes was tractable in the block code case and in part because duality theory suggests that self-dual matrix codes may attain a perfectly balanced trade-off between efficiency and error correction. Thus, we restrict to the class of self-dual matrix codes and employ a method based on double cosets of the matrix equivalence maps to enumerate the equivalence classes of these codes for small parameters.
Morrison, Katherine, "Equivalence and duality for rank-metric and matrix codes" (2012). ETD collection for University of Nebraska - Lincoln. AAI3522082.