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Source: Dual code

In coding theory, the dual code of a linear code

C
⊂

F

q

n

{\displaystyle C\subset \mathbb {F} _{q}^{n}}

is the linear code defined by

C

⊥

=
{
x
∈

F

q

n

∣
⟨
x
,
c
⟩
=
0

∀
c
∈
C
}

{\displaystyle C^{\perp }=\{x\in \mathbb {F} _{q}^{n}\mid \langle x,c\rangle =0\;\forall c\in C\}}

where

⟨
x
,
c
⟩
=

∑

i
=
1

n

x

i

c

i

{\displaystyle \langle x,c\rangle =\sum _{i=1}^{n}x_{i}c_{i}}

is a scalar product. In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form

⟨
⋅
⟩

{\displaystyle \langle \cdot \rangle }

. The dimension of C and its dual always add up to the length n:

dim
⁡
C
+
dim
⁡

C

⊥

=
n
.

{\displaystyle \dim C+\dim C^{\perp }=n.}

A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code.