Constructions of Rank-Metric Codes of Small Tensor Rank

2026-05-20

AI Summary

This study addresses the construction of rank-metric codes whose tensor rank approaches the theoretical lower bound \(k + d - 1\). To this end, the work introduces the novel concept of “tensor rank defect,” which elucidates an intrinsic connection between the tensor rank of a rank-metric code and the parameters of its associated Hamming-metric linear code. By leveraging algebraic geometry codes in conjunction with tensor decomposition techniques and analysis of matrix spaces over finite fields, the authors present the first systematic construction of rank-metric codes with small tensor rank defect. These constructions not only come close to achieving the minimum tensor rank (MTR) bound but also provide new tools for efficient coding schemes and advances in algebraic complexity theory.

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.

Problem

• rank-metric codes
• tensor rank
• minimal tensor rank
• tensor rank defect
• MDS codes

Innovation

• tensor rank defect
• rank-metric codes
• algebraic geometry codes
• MTR codes
• Kruskal bound