Orthogonal linear code

Let $C \leq 𝕂 𝑛 𝑞$ be a $[𝑛, 𝑘]$ -code. The orthogonal code¹ $C ⟂ \leq 𝕂 𝑛 𝑞$ is then a $[𝑛, 𝑛 - 𝑘]$ -code given by its orthogonal complement #m/def/code

C ⟂ = {𝑆 \in 𝕂 𝑛 𝑞 : C 𝖳 𝑆 = 0}

For a $[𝑛, 𝑛 / 2]$ -code it is possible to be orthogonal-dual, i.e. $C = C ⟂$ .²

Properties

If $𝐺 = [𝟙 𝑘 ∣ 𝑃]$ generates $C$ , then $𝐻 = [- 𝑃 𝖳 ∣ 𝟙 𝑛 - 𝑘]$ generates $C ⟂$ , and is the parity check matrix for $C$ .

Proof

Note $𝐺 𝐻 𝖳 = 0$ and $𝐻$ has correct size and rank, thus

𝑥 \in C ⟺ 𝑥 𝐻 𝖳 = ⃗ 𝟎

as required.

#state/tidy | #lang/en | #SemBr

The more popular terminology is dual code, but this is confusing. ↩
1999. Introduction to coding theory, §3.2, p. 36 ↩

Orthogonal linear code

Properties

Footnotes