A Construction of MDS Quantum Convolutional Codes

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 $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. 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 $q\ge 5$ is an odd prime power and $2\le i\le (q-1)\mathrm{/}2$; {\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 $10m+7$ ($m\ge 2$), and $2\le i\le 2m-1$.