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

## TL;DR

This paper investigates the existence of perfect codes in the sum-rank metric, establishing non-existence results for multiple-block spaces and providing bounds and conditions for their potential existence.

## Contribution

It extends the understanding of perfect codes in the sum-rank metric by analyzing geometry, deriving bounds, and proving non-existence in various cases.

## Key findings

- Non-existence of perfect codes in multiple-block sum-rank spaces for certain parameters
- Explicit bounds and constraints for perfect codes in two-block spaces
- Computational evidence supporting non-existence based on volume and divisibility conditions

## 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.