Constructions of Rank-Metric Codes of Small Tensor Rank

TL;DR

This paper explores the tensor rank of rank-metric codes, introduces tensor rank defect, and presents new algebraic geometry-based constructions for codes with small tensor rank defect.

Contribution

It establishes new relationships between tensor rank and code parameters, and develops novel constructions of rank-metric codes with small tensor rank defect using algebraic geometry codes.

Findings

Tensor rank of codes relates to associated linear code parameters.

Introduces the concept of tensor rank defect.

Provides new algebraic geometry constructions for small tensor rank defect codes.

Abstract

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.