Coding theory MOC

Hamming distance

Let 𝑆 be an arbitrary set. The Hamming distance 𝑑(𝑥,𝑦) of 𝑥,𝑦 ∈𝑆𝑛 is the number of positions in which they differ, code i.e.

𝑑(𝑥,𝑦)=𝑛∑𝑖=1[𝑥𝑖=𝑦𝑖]

where we have used an Iverson bracket. This defines a metric on 𝑆𝑛.

• A Hamming ball B𝑟⁡(𝑐) ={𝑥 ∈𝑆𝑛 :𝑑(𝑥,𝑐) ≤𝑟} is a closed ball under this metric.
• The distance of a word from a distinguished ⃗𝟎 word is called the Hamming weight, or just weight, which sees a generalization in the linear case to Generalized Hamming weight.

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