Non-minimum tensor rank Gabidulin codes

TL;DR

This paper investigates the tensor rank of certain Gabidulin codes, revealing that some are not minimum tensor rank and identifying the first infinite family with this property.

Contribution

It determines the tensor rank of specific small-dimension Gabidulin codes and introduces the first infinite family of such codes that are not minimum tensor rank.

Findings

The tensor rank of any 8-dimensional generalized Gabidulin code in f4 is determined.

Such codes are shown to never be minimum tensor rank.

First infinite family of Gabidulin codes not minimum tensor rank identified.

Abstract

The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an $8$-dimensional $\mathbb{F}_q$-linear generalized Gabidulin code in $\mathbb{F}_{q}^{4\times4}$. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank.