7.4 More QCCs from BCH Codes
183
Applying Lemmas 7.4.7 and 7.4.8 we obtain more QCCs.
Theorem 7.4.7 Let q ≥ 4 be a prime power and n = q
m
− 1. Assume that m =
ord n (q) ≥ 3. Then there exists an [(n, n − 2m(c − 1) − 2, 1; m, d f ≥ c + 2)] q
quantum convolutional code, where 2 ≤ c = i + j ≤ q − 2 and i, j ≥ 1.
Proof Left to the reader.
Exercise 7.4.3 Prove Theorem 7.4.7.
Theorem 7.4.8 Let n = q
m
− 1, where q ≥ 4 is a prime power and assume that
m = ord n (q) ≥ 3. Then there exists an [(n, n − 2mi − 2, 1; m j, d f ≥ i + j + 2)] q
QCC, where 1 ≤ i = j ≤ q − 2.
Proof Left to the reader.
Theorem 7.4.9 Suppose that n = q
m
− 1, where q ≥ 4 is a prime power and m =
or d n (q) ≥ 3. Then there exist quantum convolutional codes with parameters
(a) [(n, n − 2m(q − 2) − 2, 1; m, d f ≥ q + 2)] q ;
(b) [(n, n − 2m(q − 1), 1; m + 1, d f ≥ q + 3)] q ;
(c) [(n, n − 2m(q − 1) − 2, 1; m j, d f ≥ q + j + 2)] q , for each 1 ≤ j < q − 1;
(d) [(n, n − 2m(2q − 3), 1; m, d f ≥ 2q + 1)] q .
Exercise 7.4.4 Prove Theorems 7.4.8 and 7.4.9.
7.4.4 Code Comparison
In this section, we compare the parameters of the QCCs constructed here with the
ones available in [6]. The parameters [(n, k, μ; γ, d f )] q denote the parameters of
QCCs shown in that paper.
The parameters of our code shown in Table 7.1 are obtained from Construction I
(see Theorem 7.4.2), Construction II (see Theorem 7.4.4) and from Construction III
(see Theorem 7.4.8), respectively.
The criterion adopted to compare the codes is the usual: if the codes have the same
code length and the same lower bound for the free distance, the code with greater
dimension is better than the other. For example, our [(624, 598, 1; 12, d f ≥ 14)] 5
code is better than the [(624, 592, 1; γ, d f ≥ 14)] 5 code shown in [6] since these
two codes have same code length (624) and same lower bound for the free distance
(14), but the new code has greater dimension (598) than the dimension (592) of the
[(624, 592, 1; γ, d f ≥ 14)] 5 code. According to the criterion established, we can see
in Table 7.1 that the our codes are better than the ones shown in [6].
183
Applying Lemmas 7.4.7 and 7.4.8 we obtain more QCCs.
Theorem 7.4.7 Let q ≥ 4 be a prime power and n = q
m
− 1. Assume that m =
ord n (q) ≥ 3. Then there exists an [(n, n − 2m(c − 1) − 2, 1; m, d f ≥ c + 2)] q
quantum convolutional code, where 2 ≤ c = i + j ≤ q − 2 and i, j ≥ 1.
Proof Left to the reader.
Exercise 7.4.3 Prove Theorem 7.4.7.
Theorem 7.4.8 Let n = q
m
− 1, where q ≥ 4 is a prime power and assume that
m = ord n (q) ≥ 3. Then there exists an [(n, n − 2mi − 2, 1; m j, d f ≥ i + j + 2)] q
QCC, where 1 ≤ i = j ≤ q − 2.
Proof Left to the reader.
Theorem 7.4.9 Suppose that n = q
m
− 1, where q ≥ 4 is a prime power and m =
or d n (q) ≥ 3. Then there exist quantum convolutional codes with parameters
(a) [(n, n − 2m(q − 2) − 2, 1; m, d f ≥ q + 2)] q ;
(b) [(n, n − 2m(q − 1), 1; m + 1, d f ≥ q + 3)] q ;
(c) [(n, n − 2m(q − 1) − 2, 1; m j, d f ≥ q + j + 2)] q , for each 1 ≤ j < q − 1;
(d) [(n, n − 2m(2q − 3), 1; m, d f ≥ 2q + 1)] q .
Exercise 7.4.4 Prove Theorems 7.4.8 and 7.4.9.
7.4.4 Code Comparison
In this section, we compare the parameters of the QCCs constructed here with the
ones available in [6]. The parameters [(n, k, μ; γ, d f )] q denote the parameters of
QCCs shown in that paper.
The parameters of our code shown in Table 7.1 are obtained from Construction I
(see Theorem 7.4.2), Construction II (see Theorem 7.4.4) and from Construction III
(see Theorem 7.4.8), respectively.
The criterion adopted to compare the codes is the usual: if the codes have the same
code length and the same lower bound for the free distance, the code with greater
dimension is better than the other. For example, our [(624, 598, 1; 12, d f ≥ 14)] 5
code is better than the [(624, 592, 1; γ, d f ≥ 14)] 5 code shown in [6] since these
two codes have same code length (624) and same lower bound for the free distance
(14), but the new code has greater dimension (598) than the dimension (592) of the
[(624, 592, 1; γ, d f ≥ 14)] 5 code. According to the criterion established, we can see
in Table 7.1 that the our codes are better than the ones shown in [6].