Abstract
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 $[\![n,k{: }m,d]\!]_q$ 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
$[\![7,1{: }1,3]\!]_2$, $[\![9,2{: }2,3]\!]_2$, $[\![10,3{: }2,3]\!]_2$,
$[\![11,4{: }2,3]\!]_2$, $[\![11,1{: }2,4]\!]_2$, $[\![13,1{: }4,4]\!]_2$,
$[\![13,1{: }1,5]\!]_2$, $[\![14,1{: }2,5]\!]_2$, $[\![15,1{: }3,5]\!]_2$,
$[\![19,9{: }1,4]\!]_2$, $[\![20,9{: }2,4]\!]_2$, $[\![21,9{: }3,4]\!]_2$,
$[\![22,9{: }4,4]\!]_2$ have been found. All these codes have better parameters
than hybrid codes obtained from the best known stabilizer quantum codes.
