# Tensor Representation of Rank-Metric Codes

## TL;DR

This paper develops a tensor-based framework for rank-metric codes, introducing generator and parity check tensors, defining tensor rank as a key parameter, and exploring the existence and construction of minimal tensor rank codes related to MDS codes.

## Contribution

It introduces the tensor representation of rank-metric codes, defines tensor rank as a new invariant, and establishes connections with MDS codes, including explicit constructions of minimal tensor rank codes.

## Key findings

- Tensor rank of a code is at least k+d-1.
- Minimal tensor rank (MTR) codes meet this bound.
- Explicit constructions of MTR codes from MDS codes.

## Abstract

We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor rank of a code to be the tensor rank of its generating tensors, and propose that this quantity is a significant coding theoretic parameter. By a result on the tensor rank of Kruskal from the 1970s, the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k+d-1$. We call codes that meet this bound minimal tensor rank (MTR) codes. It is known from results in algebraic complexity theory that an MTR code implies the existence of an MDS code. In this paper, we also address the converse problem, that of the existence of an MTR code, given an MDS code. We identify several parameters for which the converse holds and give explicit constructions of MTR codes using MDS codes. We furthermore define generalized tensor ranks, which give a refinement of the tensor rank as a code invariant. Moreover, we use these to distinguish inequivalent rank-metric codes.