# Perfect codes from PGL(2,5) in Star graphs

## TL;DR

This paper constructs new perfect codes in Star graphs using the embedding of PGL(2,5) into Sym_6, expanding the known classes of perfect codes in these graphs.

## Contribution

It introduces a novel class of perfect codes in Star graphs derived from the cosets of PGL(2,5) embedded in Sym_6, for all n ≥ 6.

## Key findings

- Existence of perfect codes from PGL(2,5) in S_6 and higher
- Construction of perfect codes as unions of cosets
- Extension of known perfect code classes in Star graphs

## Abstract

The Star graph $S_n$ is the Cayley graph of the symmetric group $Sym_n$ with the generating set $\{(1\mbox{ }i): 2\leq i\leq n \}$. Arumugam and Kala proved that $\{\pi\in Sym_n: \pi(1)=1\}$ is a perfect code in $S_n$ for any $n, n\geq 3$. In this note we show that for any $n, n\geq 6$ the Star graph $S_n$ contains a perfect code which is a union of cosets of the embedding of $PGL(2,5)$ into $Sym_6$.