# perfect code

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

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

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

The covering radius of $C$ is

 $\displaystyle r(C)=\max_{x}\min_{c}\delta(x,c)$

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

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.

Title perfect code PerfectCode 2013-03-22 14:23:43 2013-03-22 14:23:43 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 11T71 packing radius covering radius