MDS Entanglement-Assisted Quantum Codes of Arbitrary Lengths and Arbitrary Distances

Quantum error correction is fundamentally important for quantum information processing and computation. Quantum error correction codes have been studied and constructed since the pioneering work of Shor and Steane. Optimal (called MDS) $q$-qubit quantum codes attaining the quantum Singleton bound were constructed for very restricted lengths $n \leq q^2+1$. Entanglement-assisted quantum error correction (EAQEC) code was proposed to use the pre-shared maximally entangled state for the purpose of enhancing error correction capability. Recently there have been a lot of constructions of such MDS EAQEC codes attaining the quantum Singleton bound for very restricted lengths. In this paper we construct such MDS EAQEC $[[n, k, d, c]]_q$ codes for arbitrary $n$ satisfying $n \leq q^2+1$ and arbitrary distance $d\leq \frac{n+2}{2}$. It is proved that for any given length $n$ satisfying $O(q^2)=n \leq q^2+1$ and any given distance $ O(q^2)=d \leq \frac{n+2}{2}$, there exist at least $O(q^2)$ MDS EAQEC $[[n, k, d, c]]_q$ codes with different $c$ parameters. Our results show that there are much more MDS entanglement-assisted quantum codes than MDS quantum codes without consumption of the maximally entangled state. This is natural from the physical point of view.