200
7 Constructions of QCCs
H =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
H 0
H
0
H 1
H
1
. . .
H μ
H
μ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
be the matrix whose rows are the vectors v i , i = 1, 2, . . . , m. The matrices H 0 , H
0 ,
H 1 , H
1 , . . . , H μ , H
μ , are mutually disjoint. The matrices H i , i = 0, 1, . . . , μ, are
chosen in such a way that
rk H i = rk H j ,
for all i, j = 0, 1, . . . , μ (the choice of the vectors in each H i is arbitrary).
In order to compute the degree of the convolutional code constructed in the
sequence, we assume that H
0 has full rank and also rk H
0 ≥ rk H
1 ≥ · · · ≥ rk H
μ .
The matrices ˜
H
i with 1 ≤ i ≤ μ are obtained from the respective matrices H
i by
adding zero-rows at the bottom such that ˜
H i has rk H
0 rows in total.
Let H be a parity check matrix of a linear block code C = [n, k, d] q , where k =
n − m. Consider the linear block code C
∗
= [n, k
∗
, d
∗
] q with parity check matrix
H
∗
=
⎡
⎢
⎢
⎢
⎣
H
0
H
1
. . .
H
μ
⎤
⎥
⎥
⎥
⎦
.
Next, we construct a matrix G 1 (D) as follows:
G 1 (D) =
⎡
⎣
H 0
−−
H
0
⎤
⎦ +
⎡
⎣
H 1
−−
˜
H
1
⎤
⎦ D +
⎡
⎣
H 2
−−
˜
H
2
⎤
⎦ D
2
+ · · · +
⎡
⎣
H μ
−−
˜
H
μ
⎤
⎦ D
μ
.
Further, let us consider the submatrices G 0 (D) and G 2 (D) of G 1 (D), given,
respectively, by
G 0 (D) = H 0 + H 1 D + H 2 D
2
+ · · · + H μ D
μ
and
G 2 (D) = H
0 + ˜
H
1 D + ˜
H
2 D
2
+ · · · + ˜
H
μ D
μ
.