# A lower bound on permutation codes of distance $n-1$

## TL;DR

This paper extends a recursive construction for permutation codes with high minimum distance, establishing a new lower bound that slightly surpasses previous guarantees derived from mutually orthogonal Latin squares.

## Contribution

It generalizes a classical recursive construction for permutation codes of distance n-1, leading to a new lower bound on their maximum size for large n.

## Key findings

- Established a lower bound M(n,n-1) ≥ n^{1.079} for large n.
- Extended recursive construction applicable to a broader class of permutation codes.
- Provided a small improvement over bounds derived from MOLS.

## Abstract

A classical recursive construction for mutually orthogonal latin squares (MOLS) is shown to hold more generally for a class of permutation codes of length $n$ and minimum distance $n-1$. When such codes of length $p+1$ are included as ingredients, we obtain a general lower bound $M(n,n-1) \ge n^{1.079}$ for large $n$, gaining a small improvement on the guarantee given from MOLS.