Lattice from a binary linear code
Properties

Binary linear code

Lattice from a binary linear code

Let $C \leq P (Ω)$ be a binary linear code and $𝕂$ be a field with [[Characteristic| $c h a r 𝕂 = 0$ ]], and let $𝕂 Ω$ be the free module so that

𝕂 Ω = ⨁ 𝑘 \in Ω 𝕂 𝛼 𝑘

which we make a quadratic space with the bilinear form

⟨ 𝛼 𝑘, 𝛼 ℓ ⟩ = 2 [𝑘 = ℓ]

where we have used an Iverson bracket, and we take $ℤ Ω \leq 𝖠 𝖻 𝕂 Ω$ in the natural way. For $𝑆 \subseteq Ω$ define

𝛼 𝑆 = \sum 𝑘 \in 𝑆 𝛼 𝑘

then the associated lattice $𝐿 C$ of $C$ is the rational lattice code

𝐿 C = \sum 𝐶 \in C ℤ 12 𝛼 𝐶 + ℤ Ω = {\sum 𝑘 \in Ω 𝑚 𝑘 𝛼 𝑘 : 𝑚 𝑘 \in 12 ℤ, {𝑘 : 𝑚 𝑘 \in ℤ + 12} \in C}

so the dual lattice is the lattice of the orthogonal code¹

𝐿 \circ C = \sum 𝐶 \in C ⟂ ℤ 12 𝛼 𝐶 + ℤ Ω = {\sum 𝑘 \in Ω 𝑚 𝑘 𝛼 𝑘 : 𝑚 𝑘 \in 12 ℤ, {𝑘 : 𝑚 𝑘 \in ℤ + 12} \in C ⟂}

Informal explanation

We start with the lattice $ℤ Ω$ and add points at half-odd coördinates corresponding to codewords. The $ℤ 2$ -linearity of $C$ is compatible with $𝐿 C$ , since $\emptyset \in C$ corresponds to the $ℤ Ω$ lattice points, and for any $𝐶, 𝐷 \in C$
$12 𝛼 𝐶 + 12 𝛼 𝐷 + ℤ Ω = 12 𝛼 𝐶 + 𝐷 + ℤ Ω$

A slight variation on this construction is the Altered lattice from a binary linear code.

Properties

$C$ is self-orthogonal of FLM type II iff $𝐿 C$ is type II unimodular.

develop | en | SemBr

Footnotes

1988. Vertex operator algebras and the Monster, §10.2, pp. 302–303 ↩

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Binary linear code
Rational lattice

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