genus of a code:

The genus of an [n,k,d] code C over GF(q) is defined by

γ (C) = n+1-k -d.

Golay codes:

The binary Golay code $ GC_{23}$ is the quadratic residue code of length 23 over $ \mathbb{F}_2$. It is a $ [23,12,7]$-code. A generating matrix for $ GC_{23}$ is

\begin{displaymath} \left( \begin {array}{ccccccccccccccccccccccc} 1&0&0&0&0... ...0&0&0&0&0&0&0&1&1&0&1&1&0&1&1&1&0&0&0 \end {array} \right). \end{displaymath}
The binary Golay code $ GC_{24}$ is the code of length 24 over $ \mathbb{F}_2$ obtained by appending onto $ GC_{23}$ a zero-sum check digit. It is a $ [24,12,8]$-code. Moreover, it is self-dual in the sense that $ GC_{24}^\perp = GC_{24}$. A generating matrix for $ GC_{24}$ is
$\displaystyle \left(\begin {array}{cccccccccccccccccccccccc} 1&0&0&0&0&0&0&0&... ...skip }0&0&0&0&0&0&0&0&0&0&0&1&1&0&1&1&0&1&1&1&0&0&0&1 \end {array} \right). $

The ternary Golay code $ GC_{11}$ is the quadratic residue code of length 11 over $ \mathbb{F}_3$. It is a $ [11,6,5]$-code. A generating matrix for $ GC_{11}$ is
\begin{displaymath} \left( \begin {array}{ccccccccccc} 1&0&0&0&0&0&1&1&1&1&1... ...ign{\medskip }0&0&0&0&0 &1&1&2&2&1&0 \end {array} \right). \end{displaymath}
The ternary Golay code $ GC_{12}$ is the code of length 12 over $ \mathbb{F}_3$ obtained by appending onto $ GC_{11}$ a zero-sum check digit. It is a $ [12,6,6]$-code. Moreover, it is self-dual in the sense that $ GC_{12}^\perp = GC_{12}$. A generating matrix for $ GC_{12}$ is
\begin{displaymath} \left( \begin {array}{cccccccccccc} 1&0&0&0&0&0&0&1&1&1&... ...n{\medskip }0&0&0&0 &0&1&1&1&2&2&1&0 \end {array} \right). \end{displaymath}