Binary linear code

Lattice from a binary linear code

Let be a binary linear code and be a field with [[Characteristic|]], and let be the free module so that

which we make a quadratic space with the bilinear form

where we have used an Iverson bracket, and we take in the natural way. For define

then the associated lattice of is the rational lattice code

so the dual lattice is the lattice of the orthogonal code¹

Informal explanation

We start with the lattice and add points at half-odd coördinates corresponding to codewords. The -linearity of is compatible with , since corresponds to the lattice points, and for any

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

Properties

• is self-orthogonal of FLM type II iff is type II unimodular.

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §10.2, pp. 302–303 ↩