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perfect code (Definition)

Let $C$ be a linear $(n,k,d)$ -code over $\mb{F}_q$ .

The packing radius of $C$ is defined to be the value \begin{align*} \rho(C)=\frac{d-1}{2}. \end{align*} The covering radius of $C$ is \begin{align*} r(C)=\max_x\min_c \delta(x,c) \end{align*}with $x\in \mb{F}_q^n$ and $c\in C$ , and where $\delta$ denotes the Hamming distance on $\mb{F}_q^n$ .

The code $C$ is said to be perfect if $r(C)=\rho(C)$ .

The list of 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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Also defines:  packing radius, covering radius
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Cross-references: binary, codes, perfect, Hamming distance

This is version 2 of perfect code, born on 2004-06-04, modified 2004-06-08.
Object id is 5892, canonical name is PerfectCode.
Accessed 4025 times total.

Classification:
AMS MSC11T71 (Number theory :: Finite fields and commutative rings :: Algebraic coding theory; cryptography)

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