Decoding Algorithms for Tensor Codes

Tensor codes are a generalisation of matrix codes. Such codes are defined as subspaces of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex>-th order 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. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup>This work has emanated from research conducted with the financial support of the European Union MSCA Doctoral Networks, (HORIZON-MSCA-2021-DN-01, Project 101072316), the French Agence Nationale de la Recherche project ANR-21-CE39-0009-BARRACUDA and by Plan France 2030 ANR-22-PETQ-0008. Link to GitHub repository with programs:https://github.com/lucienfrancois/RothTensorCodes