Two families of Entanglement-assisted quantum MDS codes from constacyclic codes

TL;DR

This paper constructs two new families of entanglement-assisted quantum MDS codes from constacyclic codes, achieving larger minimum distances with fewer entangled states, thus advancing quantum error correction capabilities.

Contribution

It introduces a novel method for constructing EAQMDS codes from constacyclic codes, reducing the entanglement needed and producing codes with superior parameters.

Findings

Some codes have minimum distance greater than q.

Many codes have larger minimum distance than existing QMDS codes.

Most constructed codes are new and not previously documented.

Abstract

Entanglement-assisted quantum error correcting codes (EAQECCs) can be derived from arbitrary classical linear codes. However, it is a very difficult task to determine the number of entangled states required. In this work, using the method of the decomposition of the defining set of constacyclic codes, we construct two families of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper bound greater than q. Some of them have much larger minimum distance than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.