Decoding Algorithms for Tensor Codes
Résumé
Tensor codes are a generalisation of matrix codes. Such codes are defined as subspaces of order-$r$ tensors for which the ambient space is endowed with the tensor-rank as a metric. A class of these codes was introduced by Roth who outlined a decoding algorithm for low tensor-rank errors for particular cases. They may be viewed as a generalisation of the well-known Delsarte-Gabidulin-Roth maximum rank distance codes.
We study a generalised class of these codes. We investigate the properties of these codes and outline decoding techniques for different metrics that leverage their tensor structure. We first consider a fibre-wise decoding approach, as each fibre of a codeword corresponds to a Gabidulin codeword. We then give a generalisation of Loidreau's decoding method that corrects errors with properties constrained by the dimensions of the slice spaces and fibre spaces. The metrics we consider are upper bounded by the tensor-rank metric, and therefore these algorithms also decode tensor-rank weight errors.
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