Semidefinite and linear programming bounds for sum-rank-metric codes and non-existence results

TL;DR

This paper introduces new upper bounds for sum-rank-metric codes using spectral and optimization methods, improving existing bounds and establishing non-existence results for certain codes.

Contribution

It develops a semidefinite programming bound that outperforms previous bounds and compares linear programming and eigenvalue bounds, revealing their equivalence in key regimes.

Findings

SDP bound outperforms existing bounds in computational experiments

Linear programming and eigenvalue bounds are shown to be equivalent in extremal regimes

Methods are used to prove non-existence of certain optimal and perfect codes

Abstract

The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to determine the maximum size of a code with prescribed minimum distance. In this paper, we derive new sharp upper bounds on the size of a sum-rank-metric code using spectral and optimization techniques, including a semidefinite programming (SDP) bound that can outperform the best existing bounds based on computational experiments. Furthermore, we compare the Delsarte linear programming (LP) bound and a recent eigenvalue LP bound, and show equivalences between them, with particular emphasis on extremal regimes of the sum-rank metric. Finally, we show how to use the several SDP, LP and eigenvalue bounds to prove non-existence results for certain optimal and perfect sum-rank metric codes. Our results suggest that the combination of spectral and optimization methods effectively captures the hybrid nature of the sum-rank metric, providing new techniques that overcome the limitations of classical coding-theoretic approaches.