Linear code

Orthogonal linear code

Let be a [[Linear code|-code]]. The orthogonal code¹ is then a -code given by its orthogonal complement code

For a -code it is possible to be orthogonal-dual, i.e. .²

Properties

1. If generates , then generates , and is the ^check for .

Proof

Note and has correct size and rank, thus

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 ↩