Linear code

Orthogonal linear code

Let C ≤𝕂𝑛𝑞 be a [[Linear code|[𝑛,𝑘]-code]]. The orthogonal code¹ C⟂ ≤𝕂𝑛𝑞 is then a [𝑛,𝑛 −𝑘]-code given by its orthogonal complement code

C⟂={𝑆∈𝕂𝑛𝑞:C𝖳𝑆=0}

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

Properties

1. If 𝐺 =[𝟙𝑘 ∣𝑃] generates C, then 𝐻 =[ −𝑃𝖳 ∣𝟙𝑛−𝑘] generates C⟂, and is the ^check for C.

Proof

Note 𝐺𝐻𝖳 =0 and 𝐻 has correct size and rank, thus

𝑥∈C⟺𝑥𝐻𝖳=⃗𝟎

as required.

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Footnotes

1. The more popular terminology is dual code, but this is confusing. ↩

2. 1999. Introduction to coding theory, §3.2, p. 36 ↩