Codes for Simultaneous Transmission of Quantum and Classical Information

We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes’. We construct hybrid codes $[\text{\hspace{0.17em}⁣}[n,k:m,d]\text{\hspace{0.17em}⁣}{]}_q$ with length $n$ and distance $d$, that simultaneously transmit $k$ qudits and $m$ symbols from a classical alphabet of size $q$. Many good codes such as $[\text{\hspace{0.17em}⁣}[7,1:1,3]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[9,2:2,3]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[10,3:2,3]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[11,4:2,3]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[11,1:2,4]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[13,1:4,4]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[13,1:1,5]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[14,1:2,5]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[15,1:3,5]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[19,9:1,4]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[20,9:2,4]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[21,9:3,4]\text{\hspace{0.17em}⁣}{]}_2$, $[\text{\hspace{0.17em}⁣}[22,9:4,4]\text{\hspace{0.17em}⁣}{]}_2$ have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.