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

2026-04-30

AI Summary

This study addresses the problem of determining the maximum cardinality of sum-rank metric codes under a prescribed minimum distance. It pioneers the systematic application of semidefinite programming (SDP) alongside Delsarte’s linear programming (LP) approach to this metric, integrating spectral analysis and eigenvalue techniques to uncover equivalences among distinct upper bounds and to effectively capture the hybrid structural properties inherent to the sum-rank metric. The derived bounds substantially improve upon existing results in the literature and are leveraged to rigorously establish the nonexistence of optimal or perfect sum-rank codes for several parameter regimes.

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.

Problem

• sum-rank metric
• code size
• minimum distance
• non-existence
• optimal codes

Innovation

• semidefinite programming
• sum-rank metric
• linear programming bound
• non-existence results
• spectral methods