Two families of Entanglement-assisted Quantum MDS Codes from cyclic Codes

TL;DR

This paper constructs two new families of entanglement-assisted quantum MDS codes from classical cyclic codes, achieving larger minimum distances than existing quantum MDS codes, thus expanding the known quantum code parameters.

Contribution

It introduces two novel families of $q$-ary EAQMDS codes derived from cyclic codes, with parameters surpassing known quantum MDS codes in minimum distance.

Findings

Constructed two families of $q$-ary EAQMDS codes with larger minimum distances.

Most of these codes have parameters not previously covered in literature.

Codes are based on classical cyclic MDS codes with specific algebraic structures.

Abstract

With entanglement-assisted (EA) formalism, arbitrary classical linear codes are allowed to transform into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, based on classical cyclic MDS codes by exploiting pre-shared maximally entangled states, we construct two families of $q$-ary entanglement-assisted quantum MDS codes $[[\frac{q^{2}+1}{a},\frac{q^{2}+1}{a}-2(d-1)+c,d;c]]$, where q is a prime power in the form of $am+l$, and $a=(l^2+1)$ or $a=\frac{(l^2+1)}{5}$. We show that all of $q$-ary EAQMDS have minimum distance upper limit much larger 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.