180
7 Constructions of QCCs
with respect to β. The new matrix is denoted by H 1 and since C 1 has dimension
n − m, H 1 has m linearly independent rows. Let V generated by G(D) = ˜
H 0 + ˜
H 1 D,
where ˜
H 0 = H 0 and ˜
H 1 is obtained from H 1 by adding zero-rows at the bottom such
that ˜
H 1 has the number of rows of H 0 in total. By construction, V has parameters
(n, m(2q
2
− 3) + 1, m; 1, d f ∗ ) q 2 . The Hermitian dual V
⊥ h of the convolutional code
V has dimension n − m(2q
2
− 3) − 1 and degree m.
From construction one has d ≥ 2q
2
+ 2, d 0 ≥ 2q
2 and d 1 ≥ 2; so, by Theorem 7.1.1 Item (b), the free distance of V
⊥ h satisfies d
⊥
f ≥ 2q
2
+ 2. Thus V
⊥ h has
parameters (n, n − m(2q
2
− 3) − 1, m; μ, d
⊥ h
f ≥ 2q
2
+ 2) q 2 . We know that V ⊂
V
⊥ h . Applying Lemma 7.3.1, there exists an [(n, n − 2m(2q
2
− 3) − 2, 1; m, d f ≥
2q
2
+2)] q QCC. The proof is complete.
Theorem 7.4.4 also generates good QCCs.
Theorem 7.4.4 Let n = q
2m
− 1, where q ≥ 4 is a prime power and m = ord n
(q
2
) ≥ 3. Then there exist quantum convolutional codes with parameters [(n, n −
2mi − 2, 1; m j, d f ≥ i + j + 2)] q , for each 1 ≤ i = j ≤ q
2
− 2.
Proof Left to the reader.
Corollary 7.4.1 Let n = q
2m
− 1, where q ≥ 4 is a prime power and m = ord n
(q
2
) ≥ 3. Then there exist convolutional stabilizer codes with parameters
(a) [(n, n − 2m(i − 1) − 2, 1; m, d f ≥ i + 2)] q , for each 1 ≤ i < q
2
− 1;
(b) [(n, n − 2m(q
2
− 2) − 2, 1; m, d f ≥ q
2
+ 2)] q ;
(c) [(n, n − 2m( j + q
2
− 2) − 2, 1; m, d f ≥ j + q
2
+ 2)] q , for each 1 ≤ j <
q
2
− 1.
Exercise 7.4.1 Show Theorem 7.4.4 and Corollary 7.4.1.
Until now we only have constructed unit-memory convolutional stabilizer codes.
However, the technique utilized here can be also applied to generate multi-memory
convolutional codes. These constructions are possible due to Lemma 7.4.5, since
such lemma provides the exact parameters of the corresponding classical block codes
utilized in the proposed construction. Let us next present the constructions of families
of multi-memory QCCs.
Theorem 7.4.5 (memory two QCCs) Let n = q
2m
− 1, where q ≥ 4 is a prime
power and m = ord n (q
2
) ≥ 3. Then there exists an [(n, n − 2m(i − 2) − 2, 2; 2m,
d f ≥ i + 2)] q QCC, for each 3 ≤ i < q
2
− 1.
Proof Let C be the BCH code generated by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i−2)
(x)M
(s+i−1)
(x)M
(s+i)
(x),
7 Constructions of QCCs
with respect to β. The new matrix is denoted by H 1 and since C 1 has dimension
n − m, H 1 has m linearly independent rows. Let V generated by G(D) = ˜
H 0 + ˜
H 1 D,
where ˜
H 0 = H 0 and ˜
H 1 is obtained from H 1 by adding zero-rows at the bottom such
that ˜
H 1 has the number of rows of H 0 in total. By construction, V has parameters
(n, m(2q
2
− 3) + 1, m; 1, d f ∗ ) q 2 . The Hermitian dual V
⊥ h of the convolutional code
V has dimension n − m(2q
2
− 3) − 1 and degree m.
From construction one has d ≥ 2q
2
+ 2, d 0 ≥ 2q
2 and d 1 ≥ 2; so, by Theorem 7.1.1 Item (b), the free distance of V
⊥ h satisfies d
⊥
f ≥ 2q
2
+ 2. Thus V
⊥ h has
parameters (n, n − m(2q
2
− 3) − 1, m; μ, d
⊥ h
f ≥ 2q
2
+ 2) q 2 . We know that V ⊂
V
⊥ h . Applying Lemma 7.3.1, there exists an [(n, n − 2m(2q
2
− 3) − 2, 1; m, d f ≥
2q
2
+2)] q QCC. The proof is complete.
Theorem 7.4.4 also generates good QCCs.
Theorem 7.4.4 Let n = q
2m
− 1, where q ≥ 4 is a prime power and m = ord n
(q
2
) ≥ 3. Then there exist quantum convolutional codes with parameters [(n, n −
2mi − 2, 1; m j, d f ≥ i + j + 2)] q , for each 1 ≤ i = j ≤ q
2
− 2.
Proof Left to the reader.
Corollary 7.4.1 Let n = q
2m
− 1, where q ≥ 4 is a prime power and m = ord n
(q
2
) ≥ 3. Then there exist convolutional stabilizer codes with parameters
(a) [(n, n − 2m(i − 1) − 2, 1; m, d f ≥ i + 2)] q , for each 1 ≤ i < q
2
− 1;
(b) [(n, n − 2m(q
2
− 2) − 2, 1; m, d f ≥ q
2
+ 2)] q ;
(c) [(n, n − 2m( j + q
2
− 2) − 2, 1; m, d f ≥ j + q
2
+ 2)] q , for each 1 ≤ j <
q
2
− 1.
Exercise 7.4.1 Show Theorem 7.4.4 and Corollary 7.4.1.
Until now we only have constructed unit-memory convolutional stabilizer codes.
However, the technique utilized here can be also applied to generate multi-memory
convolutional codes. These constructions are possible due to Lemma 7.4.5, since
such lemma provides the exact parameters of the corresponding classical block codes
utilized in the proposed construction. Let us next present the constructions of families
of multi-memory QCCs.
Theorem 7.4.5 (memory two QCCs) Let n = q
2m
− 1, where q ≥ 4 is a prime
power and m = ord n (q
2
) ≥ 3. Then there exists an [(n, n − 2m(i − 2) − 2, 2; 2m,
d f ≥ i + 2)] q QCC, for each 3 ≤ i < q
2
− 1.
Proof Let C be the BCH code generated by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i−2)
(x)M
(s+i−1)
(x)M
(s+i)
(x),