Even binary code

The even binary code $E (Ω 𝑛)$ of length $𝑛$ is given by #m/def/code

E (Ω 𝑛) = {𝑆 \subseteq Ω 𝑛 : | 𝑆 | \in 2 ℤ}

which is a quadratic space with the form

𝑞 : E (Ω 𝑛) \to ℤ 2 𝑆 \mapsto | 𝑆 | 2 + 2 ℤ

which has the polar form $𝑏 𝑞 (𝑆 1, 𝑆 2) = | 𝑆 1 \cap 𝑆 2 | + 2 ℤ$ and the radical $𝕂 2 Ω 𝑛$ .

Properties

$𝑞 (𝑆) = [| 𝑆 | \notin 4 ℤ]$ .
If $𝑛$ is even, then every self-dual code contains $Ω$ (this follows from the fact that it is the radical)

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