A Construction of MDS Quantum Convolutional Codes

TL;DR

This paper introduces two new families of maximum distance separable (MDS) quantum convolutional codes, broadening the scope of code parameters and relaxing previous assumptions, with potential applications in quantum error correction.

Contribution

The paper constructs two new classes of MDS quantum convolutional codes that generalize previous results by removing certain restrictions on the parameter q.

Findings

Two classes of MDS quantum convolutional codes with explicit parameters are constructed.

The codes cover new parameter ranges, including cases where q does not satisfy earlier conditions.

The codes achieve optimal error correction capabilities as indicated by their MDS property.

Abstract

In this paper, two new families of MDS quantum convolutional codes are constructed. The first one can be regarded as a generalization of \cite[Theorem 6.5]{GGGlinear}, in the sense that we do not assume that $q\equiv1\pmod{4}$. More specifically, we obtain two classes of MDS quantum convolutional codes with parameters: {\rm (i)}~ $[(q^2+1, q^2-4i+3,1;2,2i+2)]_q$, where $q\geq5$ is an odd prime power and $2\leq i\leq(q-1)/2$; {\rm (ii)}~ $[(\frac{q^2+1}{10},\frac{q^2+1}{10}-4i,1;2,2i+3)]_q$, where $q$ is an odd prime power with the form $q=10m+3$ or $10m+7$ ($m\geq2$), and $2\leq i\leq2m-1$.