Quantum MDS Codes over Small Fields

TL;DR

This paper explores the construction of quantum MDS codes over small fields, introducing new codes through shortening techniques and providing evidence that most lengths are achievable, including a novel family for certain dimensions.

Contribution

It presents a method to generate new quantum MDS codes by shortening existing codes and introduces a new family of codes for specific field sizes.

Findings

Most admissible lengths are achievable via shortening.

New QMDS codes of length q^2+2 for q=2^m are constructed.

Numerical evidence supports the conjecture on achievable code lengths.

Abstract

We consider quantum MDS (QMDS) codes for quantum systems of dimension $q$ with lengths up to $q^2+2$ and minimum distances up to $q+1$. We show how starting from QMDS codes of length $q^2+1$ based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide numerical evidence for our conjecture that almost all admissible lengths, from a lower bound $n_0(q,d)$ on, are achievable by shortening. Some additional codes that fill gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length $q^2+2$, where $q=2^m$, that appears to be new.