7.5 QCCs from Negacyclic Codes
185
7.5 QCCs from Negacyclic Codes
In this subsection, we utilize the class of negacyclic codes [14, 16, 18, 87] in order
to construct classical and quantum MDS convolutional codes. More specifically,
we apply the famous method proposed by Piret [130] (generalized to nonbinary
alphabets in [6]), which consists in the construction of classical convolutional codes
derived from block codes. After this, we utilize our classical convolutional codes
constructed here to obtain our optimal (MDS) quantum convolutional codes. The
interested reader can consult the paper [98] for more details.
By applying this algebraic technique, it is possible to construct families of classical
and quantum convolutional codes and not by only few codes with specific parameters,
in contrast with many works where only exhaustively computational search or even
specific codes are constructed.
Our classical convolutional MDS codes constructed have parameters
• (n, n − 2i + 1, 2; 1, 2i + 2) q 2 , where q ≡ 1 (mod 4) is a power of an odd prime,
n = q
2
+ 1 and 2 ≤ i ≤ n/2 − 1;
• (n, n − 2i + 2, 2; 1, 2i + 1) q 2 , where q is a power of an odd prime, n = (q
2
+
1)/2 and 2 ≤ i ≤ (n − 1)/2;
• (n, n − 2i + 1, 2; 1, 2i + 2) q 2 , where q ≥ 5 is a power of an odd prime, n = (q
2
+
1)/2 and 2 ≤ i ≤ (n − 1)/2 − 1.
The quantum convolutional MDS codes constructed here have parameters
• [(n, n − 4i + 2, 1; 2, 2i + 2)] q , where q ≡ 1 (mod 4) is a power of an odd prime,
n = q
2
+ 1 and 2 ≤ i ≤ (q − 1)/2;
• [(n, n − 4i + 4, 1; 2, 2i + 1)] q , where q ≥ 7 is a power of an odd prime, n =
(q
2
+ 1)/2 and 2 ≤ i ≤ (q − 1)/2.
7.5.1 Negacyclic Codes
The class of negacyclic codes [14, 16, 18, 76, 87] has been studied in the literature.
This class of codes is a particular class of a more general class of constacyclic codes
[18]. In this subsection, we review basic concepts on these codes.
As usual, we assume that the length and the cardinality of q of the field F q are relatively prime. Let us consider the quotient ring R n = F q [x]/(x
n
+ 1). A negacyclic
code is a principal ideal of R n under the usual correspondence
c = (c 0 , c 1 , . . . , c n−1 ) −→ c 0 + c 1 x + · · · + c n−1 x
n−1
mod (x
n
+ 1).
The generator polynomial g(x) of a negacyclic code C divides the polynomial
(x
n
+ 1). The roots of (x
n
+ 1) are the roots of (x
2n
− 1) which are not roots of
(x
n
− 1) in some extension field of F q 2 (since we will work with codes endowed
with the Hermitian inner product).
185
7.5 QCCs from Negacyclic Codes
In this subsection, we utilize the class of negacyclic codes [14, 16, 18, 87] in order
to construct classical and quantum MDS convolutional codes. More specifically,
we apply the famous method proposed by Piret [130] (generalized to nonbinary
alphabets in [6]), which consists in the construction of classical convolutional codes
derived from block codes. After this, we utilize our classical convolutional codes
constructed here to obtain our optimal (MDS) quantum convolutional codes. The
interested reader can consult the paper [98] for more details.
By applying this algebraic technique, it is possible to construct families of classical
and quantum convolutional codes and not by only few codes with specific parameters,
in contrast with many works where only exhaustively computational search or even
specific codes are constructed.
Our classical convolutional MDS codes constructed have parameters
• (n, n − 2i + 1, 2; 1, 2i + 2) q 2 , where q ≡ 1 (mod 4) is a power of an odd prime,
n = q
2
+ 1 and 2 ≤ i ≤ n/2 − 1;
• (n, n − 2i + 2, 2; 1, 2i + 1) q 2 , where q is a power of an odd prime, n = (q
2
+
1)/2 and 2 ≤ i ≤ (n − 1)/2;
• (n, n − 2i + 1, 2; 1, 2i + 2) q 2 , where q ≥ 5 is a power of an odd prime, n = (q
2
+
1)/2 and 2 ≤ i ≤ (n − 1)/2 − 1.
The quantum convolutional MDS codes constructed here have parameters
• [(n, n − 4i + 2, 1; 2, 2i + 2)] q , where q ≡ 1 (mod 4) is a power of an odd prime,
n = q
2
+ 1 and 2 ≤ i ≤ (q − 1)/2;
• [(n, n − 4i + 4, 1; 2, 2i + 1)] q , where q ≥ 7 is a power of an odd prime, n =
(q
2
+ 1)/2 and 2 ≤ i ≤ (q − 1)/2.
7.5.1 Negacyclic Codes
The class of negacyclic codes [14, 16, 18, 76, 87] has been studied in the literature.
This class of codes is a particular class of a more general class of constacyclic codes
[18]. In this subsection, we review basic concepts on these codes.
As usual, we assume that the length and the cardinality of q of the field F q are relatively prime. Let us consider the quotient ring R n = F q [x]/(x
n
+ 1). A negacyclic
code is a principal ideal of R n under the usual correspondence
c = (c 0 , c 1 , . . . , c n−1 ) −→ c 0 + c 1 x + · · · + c n−1 x
n−1
mod (x
n
+ 1).
The generator polynomial g(x) of a negacyclic code C divides the polynomial
(x
n
+ 1). The roots of (x
n
+ 1) are the roots of (x
2n
− 1) which are not roots of
(x
n
− 1) in some extension field of F q 2 (since we will work with codes endowed
with the Hermitian inner product).