# Some constructions of quantum MDS codes

## TL;DR

This paper presents new quantum MDS codes constructed via classical Reed-Solomon codes containing their Hermitian duals, expanding known parameters and establishing non-existence results for certain code lengths.

## Contribution

The paper introduces new quantum MDS codes with specific parameters, proves non-existence of certain classical codes, and provides the first examples of quantum MDS codes with larger minimum distances.

## Key findings

- Constructed quantum MDS codes with parameters $[ exttt{!q^2+1, q^2+3-2d, d}]_q$ for specified $d$.
- Proved non-existence of classical Hermitian dual-containing Reed-Solomon codes for certain lengths.
- Presented new quantum MDS codes with larger minimum distances, including the first for $d extgreater q+2$.

## Abstract

We construct quantum MDS codes with parameters $ [\![ q^2+1,q^2+3-2d,d ]\!] _q$ for all $d \leqslant q+1$, $d \neq q$. These codes are shown to exist by proving that there are classical generalised Reed-Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if $d\geqslant q+2$ then there is no generalised Reed-Solomon $[n,n-d+1,d]_{q^2}$ code which contains its Hermitian dual. We also construct an $ [\![ 18,0,10 ]\!] _5$ quantum MDS code, an $ [\![ 18,0,10 ]\!] _7$ quantum MDS code and a $ [\![ 14,0,8 ]\!] _5$ quantum MDS code, which are the first quantum MDS codes discovered for which $d \geqslant q+3$, apart from the $ [\![ 10,0,6 ]\!] _3$ quantum MDS code derived from Glynn's code.