# On the non-existence of perfect codes in the Niederreiter-Rosenbloom-Tsfasman metric

## TL;DR

This paper investigates the existence of perfect codes in the Niederreiter-Rosenbloom-Tsfasman metric, establishing necessary conditions and proving their non-existence in certain parameter regimes.

## Contribution

It provides new necessary conditions for perfect codes in the NRT-metric and proves their non-existence for specific parameters, expanding understanding of code limitations.

## Key findings

- If a non-trivial perfect code exists, then (r+1)(R+1) ≤ rs.
- Established a connection between perfect codes and the knapsack problem.
- Proved non-existence of perfect codes for s=R+2 when r>R.

## Abstract

In this paper we consider codes in $\mathbb{F}_q^{s\times r}$ with packing radius $R$ regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of chains with the same length) and we establish necessary condition on the parameters $s,r$ and $R$ for the existence of perfect codes. More explicitly, for $r,s\geq 2$ and $R\geq 1$ we prove that if there is a non-trivial perfect code then $(r+1)(R+1)\leq rs$. We also explore a connection to the knapsack problem and establish a correspondence between perfect codes with $r>R$ and those with $r=R$. Using this correspondence we prove the non-existence of non-trivial perfect codes also for $s=R+2$.