Coding theory MOC

Perfect code

A $𝑞$ -ary code $C \subseteq 𝑆 𝑛 𝑞$ of length $𝑛$ in alphabet $𝑆 𝑞$ is said to be a perfect $𝑒$ -error correcting code, or briefly a perfect code, iff¹ #m/def/code

it has minimum distance $2 𝑒 + 1$ ; and
for every string $𝑥 \in 𝑆 𝑛 𝑞$ there exists a unique codeword $𝑐 \in C$ with $𝑑 (𝑥, 𝑐) \leq 𝑒$ .

equivalently, $C$ has a minimum distance $2 𝑒 + 1$ and covering radius $𝑒$ . See also Quasi-perfect code.

Properties

Sphere packing condition for a perfect code
The extended code of a linear perfect code is quasi-perfect.

#state/tidy | #lang/en | #SemBr

1999. Introduction to coding theory, §3.1, p. 34 ↩