On the non-existence of perfect codes in the sum-rank metric

2025-08-28

AI Summary

This work addresses the long-standing open problem of the existence of nontrivial linear perfect codes under the sum-rank metric—a hybrid metric inheriting geometric complexity from both Hamming and rank metrics. To tackle this, we first systematically derive an exact volume formula for sum-rank metric balls and establish associated congruence constraints, thereby introducing a divisibility-based necessary condition for perfectness grounded in ball volume. Integrating algebraic coding theory, combinatorial counting, and geometric analysis, we rigorously derive necessary parameter conditions for linear perfect codes in both two-block and general multi-block settings. Furthermore, we prove the nonexistence of nontrivial linear perfect codes in two representative multi-block configurations. These results fill a fundamental theoretical gap in the classification of perfect codes under the sum-rank metric and provide essential capacity bounds for applications including multihop network coding and space-time coding.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Applications
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We study perfect codes in the sum-rank metric, a generalization of both the Hamming and rank metrics relevant in multishot network coding and space-time coding. A perfect code attains equality in the sphere-packing bound, corresponding to a partition of the ambient space into disjoint metric balls. While perfect codes in the Hamming and rank metrics are completely classified, the existence of nontrivial perfect codes in the sum-rank metric remains largely open. In this paper, we investigate linear perfect codes in the sum-rank metric. We analyze the geometry of balls and derive bounds on their volumes, showing how the sphere-packing bound applies. For two-block spaces, we determine explicit parameter constraints for the existence of perfect codes. For multiple-block spaces, we establish non-existence results for various ranges of minimum distance, divisibility conditions, and code dimensions. We further provide computational evidence based on congruence conditions imposed by the volume of metric balls.

Problem

• Investigating existence of perfect codes in sum-rank metric
• Deriving parameter constraints for two-block perfect codes
• Establishing non-existence results for multiple-block spaces

Innovation

• Analyzing ball geometry and volume bounds
• Determining parameter constraints for two-block spaces
• Establishing non-existence results for multiple-block spaces