Binary linear code

FLM code types I and II

In Vertex operator algebras and the Monster, a binary linear code C ≤P(Ω𝑛) is said to be type I iff

1. 𝑛 ∈2ℤ;
2. |𝐶| ∈2ℤ for all 𝐶 ∈C, i.e. C is an even code; and
3. Ω𝑛 ∈C

and type II iff

1. 𝑛 ∈4ℤ;
2. |𝐶| ∈4ℤ for all 𝐶 ∈C, i.e. C is an doubly even code; and
3. Ω𝑛 ∈C

It follows that such codes are subcodes of the even binary code.

Properties

1. Let 𝑛 ∈4ℤ. Then C is self-orthogonal code of type II iff C/𝕂2Ω is a (maximal) totally isotropic subspace of E(Ω)/𝕂2Ω of dimension 𝑛/2 −1; equivalently C is a (maximal) totally isotropic subspace of E(Ω) of dimension 𝑛/2.

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