Eigenvalue Bounds for Sum-Rank-Metric Codes

We consider the problem of deriving upper bounds on the parameters of sum-rank-metric codes, with focus on their dimension and block length. The sum-rank metric is a combination of the Hamming and the rank metric, and most of the available techniques to investigate it seem to be unable to fully capture its hybrid nature. In this paper, we introduce a new approach based on sum-rank-metric graphs, in which the vertices are tuples of matrices over a finite field, and where two such tuples are connected when their sum-rank distance is equal to one. We establish various structural properties of sum-rank-metric graphs and combine them with eigenvalue techniques to obtain bounds on the cardinality of sum-rank-metric codes. The bounds we derive improve on the best known bounds for several choices of the parameters. While our bounds are explicit only for small values of the minimum distance, they clearly indicate that spectral theory is able to capture the nature of the sum-rank-metric better than the currently available methods. They also allow us to establish new non-existence results for (possibly nonlinear) MSRD codes.

I. Introduction

In the past decade, sum-rank-metric codes have featured prominently in the coding theory literature, mainly in connection with multi-shot network coding; see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] among many others. In that context, sum-rank-metric codes can significantly reduce the alphabet size with respect to schemes based on ordinary rank-metric codes. Sum-rank-metric codes have been proposed for other applications as well, including space-time coding and distributed storage; see for instance [5], [16], [17], [18].