[[Coding theory MOC]]
# Binary linear code

The vector space $\mathbb{K}_{2}^n$ is equivalent to the [[powerset]] $\mathcal{P}(\Omega_{n})$ where $\Omega_{n}$ is a set with $n$ elements, and addition is given by [[symmetric difference]]. #m/def/code
Thus a **binary [[linear code]]** $\mathcal{C}$ is a $\opn{GF}(2)$-[[Vector subspace|subspace]] of $\mathcal{P}(\Omega_{n})$.[^1988]
Indeed, the main advantage of this notation is that it allows the importing of notations from set theory.

- [[FLM code types I and II]]
- [[Orthogonal code]]

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §10.1, p. 299

## Further notions

- Given a codeword $C \in \mathcal{C}$ we have  $\wt C = \abs C$.
- The natural non-singular symmetric bilinear form on $\mathcal{P}(\Omega_{n})$ becomes $\abs{S_{1}\cap S_{2}} + 2\mathbb{Z}$.
- [[Even binary code]]
- [[Lattice from a binary linear code]]

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