Xing-Ling Codes, Duals of their Subcodes, and Good Asymmetric Quantum Codes

A class of powerful $q$-ary linear polynomial codes originally proposed by Xing and Ling is deployed to construct good asymmetric quantum codes via the standard CSS construction. Our quantum codes are $q$-ary block codes that encode $k$ qudits of quantum information into $n$ qudits and correct up to $\flr{(d_{x}-1)/2}$ bit-flip errors and up to $\flr{(d_{z}-1)/2}$ phase-flip errors.. In many cases where the length $(q^{2}-q)/2 \leq n \leq (q^{2}+q)/2$ and the field size $q$ are fixed and for chosen values of $d_{x} \in \{2,3,4,5\}$ and $d_{z} \ge \delta$, where $\delta$ is the designed distance of the Xing-Ling (XL) codes, the derived pure $q$-ary asymmetric quantum CSS codes possess the best possible size given the current state of the art knowledge on the best classical linear block codes.