Linear code

Orthogonal linear code

Let be a -code. The orthogonal code1 is then a -code given by its orthogonal complement #m/def/code

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

Properties

  1. If generates , then generates , and is the parity check matrix for .
Proof

Note and has correct size and rank, thus

as required.

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

Footnotes

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

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