Xiujing Zheng’s research while affiliated with Hefei University of Technology and other places

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In recent years, Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes have been widely used to construct quantum MDS codes. In this paper, we give some sufficient conditions under which a certain system of equations over ${\mathbb {F}}_{q^2}$ has a solution over ${\mathbb {F}}_q^*$, which effectively unify similar known techniques for constructing Hermitian self-orthogonal codes. Moreover, we construct five new classes of q-ary quantum MDS codes with flexible parameters from Hermitian self-orthogonal GRS codes. Compared to the previous literature, the quantum MDS codes we construct have different lengths, or the same length but larger distances. In particular, some of the quantum MDS codes we construct have distances that can be taken to the maximum distance of the quantum MDS codes from GRS codes.

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 bits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of q2-1$q^2-1$, which are completely new and unlike all those known lengths existed before.

Assume that q is a prime power and $m\geq 2$ is a positive integer. Cyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q-1)$, and constacyclic codes over $\mathbb{F}_{q^{2m}}$ of length $n=\frac{q^{2m}-1}{\rho }$ with $\rho\mid (q+1)$ are considered in this paper, respectively. Two classes of quantum codes are derived from the images of these codes by the Hermitian construction. Compared with the previously known quantum codes, the quantum codes in our scheme have better parameters.

Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing self-orthogonal condition with the help of pre-shared entanglement between the sender and the receiver. In this paper, by using generalized Reed-Solomon codes, we construct two families of entanglement-assisted quantum error-correcting MDS (EAQMDS) codes with parameters $[[\frac{b({q^2}-1)}{a}+\frac{{q^2} - 1}{a}, \frac{b({q^2}-1)}{a}+\frac{{q^2}-1}{a}-2d+c+2,d;c]]_q$, where q is a prime power and $a| (q+1)$. Among our constructions, the EAQMDS codes have much larger minimum distance than the known EAQMDS codes with the same length and consume the same number of ebits. Moreover, some of the lengths of ours EAQMDS codes may not be divisors of $q^2\pm 1$, which are new and different from all the previously known ones.