Binary linear code

Even binary code

The even binary code E(Ω𝑛) of length 𝑛 is given by code

E(Ω𝑛)={𝑆⊆Ω𝑛:|𝑆|∈2ℤ}

which is a quadratic space with the form

𝑞:E(Ω𝑛)→ℤ2𝑆↦|𝑆|2+2ℤ

which has the polar form 𝑏𝑞(𝑆1,𝑆2) =|𝑆1∩𝑆2| +2ℤ and the radical 𝕂2Ω𝑛.

Properties

Invoking the Iverson bracket,

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

---

develop | en | SemBr