Constructions of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes

TL;DR

This paper constructs new entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes, achieving larger minimum distances and introducing novel code lengths not divisible by q^2-1.

Contribution

It introduces three classes of EAQMDS codes with improved parameters and novel lengths, expanding the known range of quantum error-correcting codes.

Findings

Minimum distances are significantly larger than existing codes.

Some code lengths are not divisors of q^2-1, a new discovery.

Codes constructed outperform known EAQMDS codes of the same length and ebits.

Abstract

By generalizing the stabilizer quantum error-correcting codes, entanglement-assisted quantum error-correcting (EAQEC) codes were introduced, which could be derived from any classical linear codes via the relaxation of self-orthogonality conditions with the aid of pre-shared entanglement between the sender and the receiver. In this paper, three classes of entanglement-assisted quantum error-correcting maximum-distance-separable (EAQMDS) codes are constructed through generalized Reed-Solomon codes. Under our constructions, the minimum distances of our EAQMDS codes are much larger than those of the known EAQMDS codes of the same lengths that consume the same number of ebits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of $q^2-1$, which are completely new and unlike all those known lengths existed before.