binary Golay codeBinaryGolayCode2004-06-04 16:59:052004-06-04 16:59:05Definitionextended binary golay code% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}The \emph{binary Golay Code} $\mc{G}_{23}$ is a perfect linear binary [23,12,7]-code with a plethora of different \PMlinkescapetext{equivalent} constructions.
\subsection*{Sample Constructions}
\begin{itemize}
\item {\bf Lexicographic Construction:} Let $v_0$ be the all-zero word in $\mb{F}_2^{23}$, and inductively define $v_j$ to be the smallest word (smallest with respect to the lexicographic ordering on $\mb{F}_2^{23}$ that differs from $v_i$ in at least 7 places for all $i<j$.
\item {\bf \PMlinkescapetext{Quadratic Residue} Construction:} $\mc{G}_{23}$ is the quadratic residue code of length 23.
\end{itemize}
The \emph{extended binary Golay Code} $\mc{G}_{24}$ is obtained by appending a zero-sum check digit to the end of every word in $\mc{G}_{23}$.
Both the binary Golay code and the extended binary Golay code have some remarkable \PMlinkescapetext{properties}.
\subsection*{Properties}
\begin{itemize}
\item $\mc{G}_{24}$ has 4096 codewords: 1 of weight 0, 759 of weight 8, 2576 of weight 12, 759 of weight 18, and 1 of weight 24.
\item The automorphism group of $\mc{G}_{24}$ is the Mathieu group $M_{24}$, one of the sporadic groups.
\item The Golay Code is used to define the Leech Lattice, one of the most efficient sphere-packings known to date.
\item The optimal strategy to the mathematical game called Mogul is to always revert the current position to one corresponding to a word of the Golay code.
\item The words of weight 8 in $\mc{G}_{24}$ form a $S(5,8,24)$ Steiner system. In fact, this property uniquely determines the code.
\end{itemize}