| Version current |
Version 1 |
| Let $C$ be a \PMlinkname{linear}{LinearCode} $(n,k,d)$-code over $\mb{F}_q$. |
Let $C$ be a linear $(n,k,d)$-code over $\mb{F}_q$. |
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| The \emph{packing radius} of $C$ is defined to be the value |
The \emph{packing radius} of $C$ is defined to be the value |
| \begin{align*} |
\begin{align*} |
| \rho(C)=\frac{d-1}{2}. |
\rho(C)=\frac{d-1}{2}. |
| \end{align*} |
\end{align*} |
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| The \emph{covering radius} of $C$ is |
The \emph{covering radius} of $C$ is |
| \begin{align*} |
\begin{align*} |
| r(C)=\max_x\min_c \delta(x,c) |
r(C)=\max_x\min_c \delta(x,c) |
| \end{align*} |
\end{align*} |
| with $x\in \mb{F}_q^n$ and $c\in C$, and where $\delta$ denotes the Hamming distance on $\mb{F}_q^n$. |
with $x\in \mb{F}_2^n$ and $c\in C$, and where $\delta$ denotes the Hamming distance on $\mb{F}_2^n$. |
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The code $C$ is said to be \emph{perfect} if $r(C)=\rho(C)$. |
| The \PMlinkname{code}{Code} $C$ is said to be \emph{perfect} if $r(C)=\rho(C)$. |
The list of \PMlinkescapetext{classes} of linear perfect codes is very short, including only trivial codes, Hamming codes (i.e. $\rho=1$), and the binary and ternary Golay codes. |
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| The list of \PMlinkescapetext{classes} of linear perfect codes is very short, including only trivial codes, Hamming codes (i.e. $\rho=1$), and the binary and ternary \PMlinkname{Golay}{BinaryGolayCode} codes. |
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