[[Binary linear code]]
# FLM code types I and II

In [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], a [[binary linear code]] $\mathcal{C} \leq \mathcal{P}(\Omega_{n})$ is said to be **type I** iff

1. $n \in 2\mathbb{Z}$;
2. $\abs C \in 2\mathbb{Z}$ for all $C \in \mathcal{C}$, i.e. $\mathcal{C}$ is an [[Divisible code|even code]]; and
3. $\Omega_{n} \in \mathcal{C}$

and **type II** iff

1. $n \in 4\mathbb{Z}$;
2. $\abs C \in 4\mathbb{Z}$ for all $C \in \mathcal{C}$, i.e. $\mathcal{C}$ is an [[Divisible code|doubly even code]]; and
3. $\Omega_{n} \in \mathcal{C}$

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

## Properties

1. Let $n \in 4\mathbb{Z}$. Then $\mathcal{C}$ is self-orthogonal code of type II iff $\mathcal{C} / \mathbb{K}_{2}\Omega$ is a (maximal) [[Quadratic space#^totallyIsotropic]] subspace of $\mathcal{E}(\Omega) / \mathbb{K}_{2}\Omega$ of dimension $n / 2 - 1$;
   equivalently $\mathcal{C}$ is a (maximal) totally isotropic subspace of $\mathcal{E}(\Omega)$ of dimension $n / 2$.

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