Coding theory MOC

Perfect code

A 𝑞-ary code C ⊆𝑆𝑛𝑞 of length 𝑛 in alphabet 𝑆𝑞 is said to be a perfect 𝑒-error correcting code, or briefly a perfect code, iff¹ code

• it has ^minimumDistance 2𝑒 +1; and
• for every string 𝑥 ∈𝑆𝑛𝑞 there exists a unique codeword 𝑐 ∈C with 𝑑(𝑥,𝑐) ≤𝑒.

equivalently, C has a minimum distance 2𝑒 +1 and ^coveringRadius 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.

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Footnotes

1. 1999. Introduction to coding theory, §3.1, p. 34 ↩