perfect code

Let $C$ be a linear (http://planetmath.org/LinearCode) $(n,k,d)$-code over ${𝔽}_q$.

The packing radius of $C$ is defined to be the value

$\rho (C)={\displaystyle \frac{d-1}{2}}.$

The covering radius of $C$ is

$r(C)=\underset{x}{\max }\underset{c}{\min }\delta (x,c)$

with $x\in {𝔽}_q^n$ and $c\in C$, and where $\delta$ denotes the Hamming distance on ${𝔽}_q^n$.

The code (http://planetmath.org/Code) $C$ is said to be perfect if $r(C)=\rho (C)$.

The list of of linear perfect codes is very short, including only trivial codes, Hamming codes (i.e. $\rho =1$), and the binary and ternary Golay (http://planetmath.org/BinaryGolayCode) codes.