7.1 Convolutional Codes
165
Definition 7.1.5 The Hermitian inner product is defined as u(D) | v(D) h =
i u i ·
v
q
i , where u i , v i ∈ F
n
q 2 and v
q
i = (v
q
1i , . . . , v
q
ni ). The Hermitian dual of a code C is
defined by
C
⊥ h = {u(D) ∈ F q 2 [D]
n
| |u(D) | v(D) h = 0 ∀ v(D) ∈ C}.
We will now show how to construct a convolutional code by means of a linear
block code. This technique is due to Piret [130]. The initial point is to consider
C ⊆ F
n
q as an [n, k, d] q linear block code with parity check matrix H . In order to
construct a convolutional code derived from C, we proceed as follows:
• Split H into μ + 1 disjoint submatrices H i such that H =
⎡
⎢
⎢
⎢
⎣
H 0
H 1
. . .
H μ
⎤
⎥
⎥
⎥
⎦
, where each
H i has n columns.
• Construct the polynomial matrix G(D) = ˜
H 0 + ˜
H 1 D + ˜
H 2 D
2
+ · · · + ˜
H μ D
μ ,
where ˜
H i , for all 1 ≤ i ≤ μ, are derived from the respective matrices H i by adding
zero-rows at the bottom such that ˜
H i has κ rows in total, where κ is the maximal
number of rows among the matrices H i .
• The matrix G(D) generates a convolutional code V .
The following theorem due to Aly et al. is a powerful tool in order to construct
convolutional codes. This method is, in fact, a generalization of Piret’s technique
[130] to nonbinary alphabets.
Theorem 7.1.1 ([6, Theorem 3]) Let C ⊆ F
n
q be a linear code with parameters
[n, k, d] q . Assume that H ∈ F
(n−k)×n
q
is a parity check matrix for C partitioned
into submatrices H 0 , H 1 , . . . , H μ as above such that κ = rk H 0 and rk H i ≤ κ for
1 ≤ i ≤ μ.
(a) The matrix G(D) is a reduced basic generator matrix.
(b) If d f and d
⊥
f denote the free distances of V and V
⊥ , respectively, d i denote the
minimum distance of the code C i = {v ∈ F
n
q | v ˜
H
t
i = 0} and d
⊥ is the minimum
distance of C
⊥ , then one has min{d 0 + d μ , d} ≤ d
⊥
f ≤ d and d f ≥ d
⊥ .
(c) If C contains its Euclidean dual C
⊥ , respectively, its Hermitian dual C
⊥ h , then the
convolutional code V = {v(D) = u(D)G(D) | u(D) ∈ F
n−k
q
[D]} is contained
in its dual V
⊥ , respectively, its Hermitian dual V
⊥ h .
7.2 Defining QCCs
Researches dealing with constructions of quantum convolutional codes (QCCs) have
received few attention when compared with its classical counterpart [5–7, 32, 41, 57–
59, 66, 98, 99, 123, 124, 154, 160, 161, 170]. This is natural because the quantum
165
Definition 7.1.5 The Hermitian inner product is defined as u(D) | v(D) h =
i u i ·
v
q
i , where u i , v i ∈ F
n
q 2 and v
q
i = (v
q
1i , . . . , v
q
ni ). The Hermitian dual of a code C is
defined by
C
⊥ h = {u(D) ∈ F q 2 [D]
n
| |u(D) | v(D) h = 0 ∀ v(D) ∈ C}.
We will now show how to construct a convolutional code by means of a linear
block code. This technique is due to Piret [130]. The initial point is to consider
C ⊆ F
n
q as an [n, k, d] q linear block code with parity check matrix H . In order to
construct a convolutional code derived from C, we proceed as follows:
• Split H into μ + 1 disjoint submatrices H i such that H =
⎡
⎢
⎢
⎢
⎣
H 0
H 1
. . .
H μ
⎤
⎥
⎥
⎥
⎦
, where each
H i has n columns.
• Construct the polynomial matrix G(D) = ˜
H 0 + ˜
H 1 D + ˜
H 2 D
2
+ · · · + ˜
H μ D
μ ,
where ˜
H i , for all 1 ≤ i ≤ μ, are derived from the respective matrices H i by adding
zero-rows at the bottom such that ˜
H i has κ rows in total, where κ is the maximal
number of rows among the matrices H i .
• The matrix G(D) generates a convolutional code V .
The following theorem due to Aly et al. is a powerful tool in order to construct
convolutional codes. This method is, in fact, a generalization of Piret’s technique
[130] to nonbinary alphabets.
Theorem 7.1.1 ([6, Theorem 3]) Let C ⊆ F
n
q be a linear code with parameters
[n, k, d] q . Assume that H ∈ F
(n−k)×n
q
is a parity check matrix for C partitioned
into submatrices H 0 , H 1 , . . . , H μ as above such that κ = rk H 0 and rk H i ≤ κ for
1 ≤ i ≤ μ.
(a) The matrix G(D) is a reduced basic generator matrix.
(b) If d f and d
⊥
f denote the free distances of V and V
⊥ , respectively, d i denote the
minimum distance of the code C i = {v ∈ F
n
q | v ˜
H
t
i = 0} and d
⊥ is the minimum
distance of C
⊥ , then one has min{d 0 + d μ , d} ≤ d
⊥
f ≤ d and d f ≥ d
⊥ .
(c) If C contains its Euclidean dual C
⊥ , respectively, its Hermitian dual C
⊥ h , then the
convolutional code V = {v(D) = u(D)G(D) | u(D) ∈ F
n−k
q
[D]} is contained
in its dual V
⊥ , respectively, its Hermitian dual V
⊥ h .
7.2 Defining QCCs
Researches dealing with constructions of quantum convolutional codes (QCCs) have
received few attention when compared with its classical counterpart [5–7, 32, 41, 57–
59, 66, 98, 99, 123, 124, 154, 160, 161, 170]. This is natural because the quantum