Bilinear Complexity of 3-Tensors Linked to Coding Theory

TL;DR

This paper investigates the tensor rank of 3-tensors, deriving upper bounds and explicit constructions, and applies these results to coding theory to analyze the complexity of certain rank-metric codes.

Contribution

It provides new upper bounds and explicit constructions for the tensor ranks of specific 3-tensors, linking algebraic complexity to coding theory.

Findings

Derived upper bounds on tensor ranks of certain 3-tensors

Constructed explicit sets of rank one matrices containing slice spaces

Showed some codes are extremal with respect to Kruskal's bound

Abstract

A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of non-scalar multiplications required to evaluate it. This quantity is also described as its tensor rank, which is the smallest number of rank one matrices whose span contains its first slice space. In this paper we derive upper bounds on the tensor ranks of certain classes of $3$-tensors and give explicit constructions of sets of rank one matrices containing their first slice spaces. We also show how these results can be applied in coding theory to derive upper bounds on the tensor rank of some rank-metric codes. In particular, we compute the tensor rank of some families of $\mathbb{F}_{q^m}$-linear codes and we show that they are extremal with respect to Kruskal's tensor rank bound.