190
7 Constructions of QCCs
7.5.3 More Optimal QCCs
We propose in this subsection the construction of MDS convolutional stabilizer codes
derived from the convolutional codes constructed in Sect. 7.5.2. We begin by recalling
some results that we will need in our constructions.
Lemma 7.5.3 ([77]) Let n = q
2
+ 1, where q ≡ 1 (mod 4) is a power of an odd
prime and suppose that s = n/2. If C is a q
2 -ary negacyclic code of length n with
defining set Z = ∪
δ
i=0 C s−2i , where 0 ≤ δ ≤ (q − 1)/2, then C
⊥ h ⊆ C.
Lemma 7.5.4 ([77]) Let n = (q
2
+ 1)/2, where q is a power of an odd prime. If
C is a q
2 -ary negacyclic code of length n with defining set Z = ∪
δ
i=1 C 2i−1 , where
1 ≤ δ ≤ (q − 1)/2, then C
⊥ h ⊆ C.
Keeping these results in mind, we are able to show how to obtain families of
optimal quantum convolutional codes.
Theorem 7.5.5 Let n = q
2
+ 1, where q ≡ 1 (mod 4) is a power of an odd prime
and suppose that s = n/2. Then there exists a quantum MDS convolutional code
with parameters [(n, n − 4i + 2, 1; 2, 2i + 2)] q , where 2 ≤ i ≤ (q − 1)/2.
Proof We consider the same notation utilized in Theorem 7.5.2. From Theorem 7.5.2, there exists a classical convolutional MDS code V
⊥ h with parameters
(n, n − 2i + 1, 2; 1, 2i + 2) q 2 , for each 2 ≤ i ≤ n/2 − 1. This code is the Hermitian dual of the code V with parameters (n, 2i − 1, 2; 1, d f ) q 2 . From Lemma 7.5.3
and from Theorem 7.1.1 Item (b), one has V ⊂ V
⊥ h . Applying Lemma 7.3.1, there
exists an [(n, n − 4i + 2, 1; 2, d f ≥ 2i + 2)] q convolutional stabilizer code Q, for
each 2 ≤ i ≤ (q − 1)/2. Replacing the parameters of Q in Theorem 7.3.1, the result
follows.
Theorem 7.5.6 Let n = (q
2
+ 1)/2, where q ≥ 7 is a power of an odd prime.
Then there exists a quantum MDS convolutional code with parameters [(n, n − 4i +
4, 1; 2, 2i + 1)] q , where 2 ≤ i ≤ (q − 1)/2.
Proof Left to the reader.
Exercise 7.5.2 Prove Theorem 7.5.6.
Table 7.2 given in the sequence, contains the parameters of some optimal quantum
convolutional codes constructed here.
7.6 QCCs from AG Codes
This subsection is devoted to the construction of families of quantum convolutional
codes derived from algebraic geometry codes. All the results presented here can be
found in our paper [127].
7 Constructions of QCCs
7.5.3 More Optimal QCCs
We propose in this subsection the construction of MDS convolutional stabilizer codes
derived from the convolutional codes constructed in Sect. 7.5.2. We begin by recalling
some results that we will need in our constructions.
Lemma 7.5.3 ([77]) Let n = q
2
+ 1, where q ≡ 1 (mod 4) is a power of an odd
prime and suppose that s = n/2. If C is a q
2 -ary negacyclic code of length n with
defining set Z = ∪
δ
i=0 C s−2i , where 0 ≤ δ ≤ (q − 1)/2, then C
⊥ h ⊆ C.
Lemma 7.5.4 ([77]) Let n = (q
2
+ 1)/2, where q is a power of an odd prime. If
C is a q
2 -ary negacyclic code of length n with defining set Z = ∪
δ
i=1 C 2i−1 , where
1 ≤ δ ≤ (q − 1)/2, then C
⊥ h ⊆ C.
Keeping these results in mind, we are able to show how to obtain families of
optimal quantum convolutional codes.
Theorem 7.5.5 Let n = q
2
+ 1, where q ≡ 1 (mod 4) is a power of an odd prime
and suppose that s = n/2. Then there exists a quantum MDS convolutional code
with parameters [(n, n − 4i + 2, 1; 2, 2i + 2)] q , where 2 ≤ i ≤ (q − 1)/2.
Proof We consider the same notation utilized in Theorem 7.5.2. From Theorem 7.5.2, there exists a classical convolutional MDS code V
⊥ h with parameters
(n, n − 2i + 1, 2; 1, 2i + 2) q 2 , for each 2 ≤ i ≤ n/2 − 1. This code is the Hermitian dual of the code V with parameters (n, 2i − 1, 2; 1, d f ) q 2 . From Lemma 7.5.3
and from Theorem 7.1.1 Item (b), one has V ⊂ V
⊥ h . Applying Lemma 7.3.1, there
exists an [(n, n − 4i + 2, 1; 2, d f ≥ 2i + 2)] q convolutional stabilizer code Q, for
each 2 ≤ i ≤ (q − 1)/2. Replacing the parameters of Q in Theorem 7.3.1, the result
follows.
Theorem 7.5.6 Let n = (q
2
+ 1)/2, where q ≥ 7 is a power of an odd prime.
Then there exists a quantum MDS convolutional code with parameters [(n, n − 4i +
4, 1; 2, 2i + 1)] q , where 2 ≤ i ≤ (q − 1)/2.
Proof Left to the reader.
Exercise 7.5.2 Prove Theorem 7.5.6.
Table 7.2 given in the sequence, contains the parameters of some optimal quantum
convolutional codes constructed here.
7.6 QCCs from AG Codes
This subsection is devoted to the construction of families of quantum convolutional
codes derived from algebraic geometry codes. All the results presented here can be
found in our paper [127].