The connections between graph theory 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. The space of tensors over a finite field, endowed with the (tensor) rank as a metric, fits into the same framework: this space can be seen as the Cayley graph generated by rank-one tensors. However, this graph is not distance-regular as soon as the tensor order is at least 3 and hence Delsarte's association-scheme approach does not directly apply to the study of the maximum size of tensor codes, that is subsets of the aforementioned metric space. 

We show that the eigenvalue method yields valuable results in this setting, sidestepping several computationally hard problems in tensor algebra, and in particular establishes the non-existence of optimal tensor codes under certain parameters. We prove that the spectrum of this graph has a recursive expression and that it depends on the possible intersections between tensor subspaces of large enough dimension and the Segre variety. We then derive ratio-type bounds on the maximum cardinality of a tensor code of given minimum distance that improve the Singleton-like bound and the sphere-packing bound for certain parameters.

In this talk, we will introduce tensor codes, their connections with other families of well-studied codes, and the hard problems arising in their study. We will then present the different expressions for the spectrum of this graph and the eigenvalue method in this context. We will finally present the resulting bounds and provide other applications of these results.