The Multilinear Forms Cayley Graph and the Eigenvalue Method for Tensor Codes

Jul 16, 2026

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Connections between graph theory, association schemes, and coding theory are built on Delsarte’s framework for the Hamming and rank metrics, where distance-regular Cayley graphs yield tractable spectral information and code bounds. This work generalizes the construction to tensors over a finite field using tensor-rank as a metric. The resulting Cayley graph from rank-one tensors is not distance-regular for order m≥3, yet its spectrum satisfies a recursive description tied to tensor subspace intersections and the Segre variety. Complete spectra for 2×3×3 tensors support eigenvalue-method bounds on tensor-code dimensions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages42

Reading Time1 h 46 min

Introduction

Background: graphs, association schemes, and coding theory

Tensor-rank metric and tensor codes

Multilinear forms Cayley graph

Motivation and focus on eigenvalues

arXiv :2607 .03479v1 [math .CO] 3 Jul 2026 The multilinear forms Cayley graph and the eigenvalue method for tensor codes. Eimear Byrne∗, Lucien François∗ Abstract The connections between graph theory, and more generally association schemes, and coding theory were established by Delsarte for the Hamming metric and rank-metric codes. The ambient metric space of Hamming-metric codes and rank-metric codes can be seen as Cayley graphs generated by words of weight one. The metrics considered then coincide with the geodesic distances of these distance-regular graphs. We focus on a generalisation of this framework to the space of tensors over a finite field, endowed with the tensor-rank as a metric. This space corresponds to the Cayley graph generated by rank-one tensors, which is not distance-regular for tensors of order at least 3. We show that the spectrum of this graph has a recursive expression and depends on the possible intersections between tensor subspaces of large enough dimension and the Segre variety. The spectrum of this graph for 3-order tensors can be expressed with the rank distribution of the rank-metric codes generated by these tensors. In particular, we obtain the complete spectrum of the graph for 2 × 3 × 3 tensors over any finite field. We apply this result to derive bounds on the dimension of tensor codes in the tensor-rank metric using the eigenvalue method, and in particular the ratio-type bound. Keywords. Tensors, multilinear forms, tensor codes, Cayley graphs, eigenvalue method. MSC codes. 11T71, 15A69, 94C15 . 1 Introduction The application of graph theory and algebraic combinatorics to coding theory has been studied extensively throughout the past five decades. Delsarte laid the foundation of theory of association schemes in coding theory with the introduction of the Hamming scheme and the bilinear forms scheme [19, 20, 21] . The fact that the scheme induced by a given metric is an association scheme relies on the distance-regularity of the distance graph of the ambient space. In both the Hamming and rank-metric case, the invariants of the corresponding association schemes yield MacWilliams identities. This was applied to produce a linear programming bound on the size of a code for a given minimum distance in the Hamming metric case. The eigenvalue method provides another approach to obtain similar results without the distance-regularity assumption, and has been applied to codes for the sum-rank metric, the Lee metric, and to alternating rankmetric codes; see [6] and the references therein. In this approach, elements of the ambient space V of a code over a finite field can be seen as the vertices of a graph for which the geodesic distance coincides with the distance endowed on V. This point of view allows one to obtain upper bounds on the size of a code of a given minimum distance using the known bounds on the independence numbers of the graph such as the the inertia-type and ratio-type bounds [2] . There also exist linear programming and mixed-integer linear programming bounds on the independence numbers using the spectrum of the graph with the aforementioned bounds [1, 5, 6] . The tensor-rank of an m-order tensor is a generalisation of the rank of a matrix and measures the algebraic complexity of the multilinear map associated to the tensor [12] . The tensor rank of a tensor x is the least number of simple tensors whose sum is equal to x. This determines a distance function on the space of ∗ School of Mathematics and Statistics, University College Dublin, [ebyrne@ucd.ie](ebyrne@ucd.ie) and lucien.francois@ucd .ie all m-order tensors whereby the tensor-rank distance between a pair of tensors is the tensor rank of their difference. Tensor codes were introduced in [40] as generalisation of rank-metric codes, motivated in part by applications to criss-cross error-correction. They are subspaces or subsets of the vector-space of m-order tensors that has been endowed with the tensor r

FAQs

What problem does the paper address regarding tensor codes?

It studies the eigenvalues of the multilinear forms Cayley graph associated with tensor-rank distance, aiming to derive bounds on the dimensions of tensor codes in the tensor-rank metric.

How is the Cayley graph constructed in this tensor setting?

Vertices correspond to m-order tensors over a finite field, and two tensors are connected when their tensor-rank distance equals 1, meaning the graph is generated by rank-one tensors (the Segre variety).

Why does Delsarte’s distance-regularity approach fail for higher-order tensors?

For tensor order m≥3, the generated Cayley graph is not distance-regular, so Delsarte’s method does not apply directly, motivating an eigenvalue-method treatment instead.

Jul 16, 2026

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