Hamming distance

Let $𝑆$ be an arbitrary set. The Hamming distance $𝑑 (𝑥, 𝑦)$ of $𝑥, 𝑦 \in 𝑆 𝑛$ is the number of positions in which they differ, #m/def/code i.e.

𝑑 (𝑥, 𝑦) = 𝑛 \sum 𝑖 = 1 [𝑥 𝑖 = 𝑦 𝑖]

where we have used an Iverson bracket. This defines a metric on $𝑆 𝑛$ .

A Hamming ball $B 𝑟 (𝑐) = {𝑥 \in 𝑆 𝑛 : 𝑑 (𝑥, 𝑐) \leq 𝑟}$ 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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