Decoding Algorithms for Tensor Codes

TL;DR

This paper explores decoding algorithms for tensor codes, generalizing existing methods to correct low tensor-rank errors by leveraging their tensor structure and properties.

Contribution

It introduces generalized decoding techniques for tensor codes, extending previous algorithms to handle errors constrained by tensor and slice space dimensions.

Findings

Decoding algorithms can correct errors with properties constrained by tensor and slice space dimensions.

Fibre-wise decoding leverages Gabidulin code structure within tensor codes.

Generalized Loidreau-Overbeck method extends error correction capabilities.

Abstract

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 also outlined a decoding algorithm for low tensor-rank errors that can be generalised to an algorithm with exponential complexity in the decoding radius. 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 their properties 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-Overbeck's decoding method that corrects errors with properties constrained by the dimensions of the slice spaces and fibre spaces. The metrics we consider are bounded from above by the tensor-rank metric, and therefore these algorithms also decode tensor-rank weight errors.