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7 Constructions of QCCs
Definition 7.1.1 A polynomial encoder matrix G(D) ∈ F q [D]
k×n is called basic if
G(D) has a polynomial right inverse. A basic generator matrix is called reduced (or
minimal), if the overall constraint length γ =
k
i=1
γ i , where γ i = max 1≤ j≤n {deg g i j },
has the smallest value among all basic generator matrices. In this case, we say that
γ is the degree of the code.
Definition 7.1.1 gives us the ingredients to define formally what we know by a
convolutional code.
Definition 7.1.2 A rate k/n convolutional code C with parameters (n, k, γ; μ, d f ) q
is a submodule of F q [D]
n generated by a reduced basic matrix G(D) = (g i j ) ∈
F q [D]
k×n , i.e., C = {u(D)G(D)|u(D) ∈ F q [D]
k
}, where n is the length, k is the
dimension, γ =
k
i=1
γ i is the degree, μ = max 1≤i≤k {γ i } is the memory and d f =
wt(C) = min{wt (v(D)) | v(D) ∈ C, v(D) = 0} is the free distance of the code.
In Definition 7.1.2, the weight of an element v(D) ∈ F q [D]
n is defined as
wt(v(D)) =
n
i=1
wt(v i (D)),
where wt(v i (D)) is the number of nonzero coefficients of v i (D). In the field of
Laurent series F q ((D)), whose elements are given by u(D) =
i u i D
i , where u i ∈
F q and u i = 0 for i ≤ r , for some r ∈ Z, we define the weight of u(D) as
wt(u(D)) =
Z
wt(u i ).
Definition 7.1.3 A generator matrix G(D) is called catastrophic if there exists a
u(D)
k
∈ F q ((D))
k of infinite Hamming weight such that u(D)
k G(D) has finite
Hamming weight.
We need to define the Euclidean dual of a convolutional code. To do this, we need
to define what we mean by Euclidean inner products of polynomials.
Definition 7.1.4 The Euclidean inner product of two n-tuples u(D) =
i u i D
i and
v(D) =
j u j D
j in F q [D]
n is defined as u(D) | v(D) =
i u i · v i . If C is a convolutional code, then its Euclidean dual code C
⊥ is defined by
C
⊥
= {u(D) ∈ F q [D]
n
| |u(D) | v(D) = 0, ∀ v(D) ∈ C}.
Similarly, we define here the Hermitian dual of a convolutional code.
7 Constructions of QCCs
Definition 7.1.1 A polynomial encoder matrix G(D) ∈ F q [D]
k×n is called basic if
G(D) has a polynomial right inverse. A basic generator matrix is called reduced (or
minimal), if the overall constraint length γ =
k
i=1
γ i , where γ i = max 1≤ j≤n {deg g i j },
has the smallest value among all basic generator matrices. In this case, we say that
γ is the degree of the code.
Definition 7.1.1 gives us the ingredients to define formally what we know by a
convolutional code.
Definition 7.1.2 A rate k/n convolutional code C with parameters (n, k, γ; μ, d f ) q
is a submodule of F q [D]
n generated by a reduced basic matrix G(D) = (g i j ) ∈
F q [D]
k×n , i.e., C = {u(D)G(D)|u(D) ∈ F q [D]
k
}, where n is the length, k is the
dimension, γ =
k
i=1
γ i is the degree, μ = max 1≤i≤k {γ i } is the memory and d f =
wt(C) = min{wt (v(D)) | v(D) ∈ C, v(D) = 0} is the free distance of the code.
In Definition 7.1.2, the weight of an element v(D) ∈ F q [D]
n is defined as
wt(v(D)) =
n
i=1
wt(v i (D)),
where wt(v i (D)) is the number of nonzero coefficients of v i (D). In the field of
Laurent series F q ((D)), whose elements are given by u(D) =
i u i D
i , where u i ∈
F q and u i = 0 for i ≤ r , for some r ∈ Z, we define the weight of u(D) as
wt(u(D)) =
Z
wt(u i ).
Definition 7.1.3 A generator matrix G(D) is called catastrophic if there exists a
u(D)
k
∈ F q ((D))
k of infinite Hamming weight such that u(D)
k G(D) has finite
Hamming weight.
We need to define the Euclidean dual of a convolutional code. To do this, we need
to define what we mean by Euclidean inner products of polynomials.
Definition 7.1.4 The Euclidean inner product of two n-tuples u(D) =
i u i D
i and
v(D) =
j u j D
j in F q [D]
n is defined as u(D) | v(D) =
i u i · v i . If C is a convolutional code, then its Euclidean dual code C
⊥ is defined by
C
⊥
= {u(D) ∈ F q [D]
n
| |u(D) | v(D) = 0, ∀ v(D) ∈ C}.
Similarly, we define here the Hermitian dual of a convolutional code.