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4 Linear Block Codes
4.2 Dual Codes
Let C be an [n, k, d] q linear code over F q with parity check matrix H . Since the
rows of H are linearly independent, H is a generator matrix of some code, called
Euclidean dual code of C, denoted by C
⊥ . The dual code C
⊥ has length n and
dimension n − k.
The dual code can be also defined by an alternative way by means of the
usual (Euclidean) inner product on F
n
q in the following way. Recall that if v =
(v 1 , v 2 , . . . , v n ) and w = (w 1 , w 2 , . . . , w n ) are two vectors in F
n
q , then the Euclidean
inner product v · w of v and w is defined as
v · w =
n
i=1
v i w i .
Based on Definition 4.1.5, we have the following.
Definition 4.2.1 The Euclidean dual code C
⊥ of o linear code C is defined as
C
⊥
= {v ∈ F
n
q |v · c = 0 ∀ c ∈ C}.
It is easy to see that if G and H are generator and parity check matrices, respectively,
for a given code C, then it follows that H and G are generator and parity check
matrices, respectively, for the dual C
⊥ .
Let C ⊆ F
n
q 2 be a linear code defined over F q 2 . In this case, we can also define
the dual code C
⊥ H of C with respect to the Hermitian inner product. To do this, let
v, w ∈ F
n
q 2 be two vectors.
Definition 4.2.2 The Hermitian inner product v|w H of v, w ∈ F
n
q 2 is defined as
v|w H = v
q
· w =
n
i=1
v
q
i w i ,
where v
q
= (v
q
1 , v
q
2 , . . . , v
q
n ).
Based on the Hermitian inner product, one has the Hermitian dual code C
⊥ H of
C.
Definition 4.2.3 Let C ⊆ F
n
q 2 be a linear code. The Hermitian dual code C
⊥ H of C
is defined by
C
⊥ H = {v ∈ F
n
q 2 |v
q
· c = 0 ∀ c ∈ C}.