# New Lower Bounds for Permutation Codes using Linear Block Codes

## TL;DR

This paper establishes new lower bounds for permutation codes by leveraging linear block code theory, specifically using parity check matrices to construct permutation codes with improved size and minimum distance.

## Contribution

It introduces a novel method connecting linear block codes to permutation codes, resulting in enhanced lower bounds and asymptotic improvements.

## Key findings

- New lower bounds for permutation codes established
- Technique uses parity check matrices of linear codes
- Asymptotic improvements in certain code regimes

## Abstract

In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an $[n,k,d]_q$ linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree $n$, having minimum distance at least $d$ and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code.