Constructions of Rank-Metric Codes of Small Tensor Rank
For coding theorists, this work provides new constructions and theoretical insights into tensor rank of rank-metric codes, though the results are incremental.
The paper establishes new relations between tensor rank of rank-metric codes and parameters of associated Hamming metric codes, and constructs rank-metric codes with small tensor rank defect using algebraic geometry codes.
Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one tensors. Kruskal showed that the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k + d - 1$, and codes meeting this bound with equality are called minimal tensor rank (MTR) codes. It is known from algebraic complexity theory that the existence of an MTR code implies the existence of a maximum distance separable (MDS) code. In this work, we establish new results relating the tensor rank of a rank-metric code to the parameters of associated linear codes in the Hamming metric and introduce the notion of tensor rank defect. We then develop new constructions of rank-metric codes with small tensor rank defect using algebraic geometry (AG) codes.