7.4 More QCCs from BCH Codes
173
7.4 More QCCs from BCH Codes
We describe in this part how to obtain families of unit-memory as well as multimemory convolutional stabilizer codes with good parameters. The content presented
here can be found in the paper [93]. The technique of construction employed in the
sequence is similar as the utilized in Sect. 7.3. Our constructions differ from those
shown given in [6] at least in two aspects: (1) we construct unit-memory and also
multi-memory QCCs whereas in [6] only unit-memory QCCs were constructed; (2)
we make use directly of minimal polynomials to define classical BCH codes utilized
in our quantum code constructions.
The first construction generates quantum convolutional codes of length n = q
4
−
1, (q ≥ 3 is a prime power), with parameters
• [(n, n − 4(i − 2) − 2, 1; 2, d f ≥ i + 1)] q , 3 ≤ i ≤ q
2
− 1;
• [(n, n − 4i − 2, 1; 2 j, d f ≥ i + j + 2)] q , 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2.
In the second, we show how to obtain quantum convolutional codes of length
n = q
2m
− 1, where q ≥ 4 is a prime power and m = or d n (q
2
) ≥ 3, derived from
the Hermitian construction. Our codes have parameters:
• [(n, n − 2m(2q
2
− 3) − 2, 1; m, d f ≥ 2q
2
+ 2)] q ;
• [(n, n − 2mi − 2, 1; m j, d f ≥ i + j + 2)] q , where 1 ≤ i = j ≤ q
2
− 2;
• [(n, n − 2m(i − 1) − 2, 1; m, d f ≥ i + 2)] q , where 1 ≤ i < q
2
− 1;
• [(n, n − 2m(q
2
− 2) − 2, 1; m, d f ≥ q
2
+ 2)] q ;
• [(n, n − 2m(i + q
2
− 2) − 2, 1; m, d f ≥ i + q
2
+ 2)] q , where 1 ≤ i < q
2
− 1;
• [(n, n − 2m(i − 2) − 2, 2; 2m, d f ≥ i + 2)] q , where 3 ≤ i < q
2
− 1;
• [(n, n − 2m(i − μ) − 2, μ; mμ, d f ≥ i − μ + 4)] q , where μ ≥ 3 and μ + 1 ≤
i < q
2
− 1.
Finally, the third construction proposed provides convolutional stabilizer codes of
length n = q
m
− 1, (q ≥ 4 is a prime power and m = or d n (q) ≥ 3), derived from
the Euclidean construction, with parameters
• [(n, n − 2m(c − 1) − 2, 1; m, d f ≥ c + 2)] q , where 2 ≤ c = i + j ≤ q − 2 and
i, j ≥ 1;
• [(n, n − 2mi − 2, 1; m j, d f ≥ i + j + 2)] q , where 1 ≤ i = j ≤ q − 2;
• [(n, n − 2m(q − 2) − 2, 1; m, d f ≥ q + 2)] q ;
• [(n, n − 2m(q − 1), 1; m + 1, d f ≥ q + 3)] q ;
• [(n, n − 2m(q − 1) − 2, 1; m j, d f ≥ q + j + 2)] q , where 1 ≤ j < q − 1;
• [(n, n − 2m(2q − 3), 1; m, d f ≥ 2q + 1)] q .
As we can see above, as families of unit-memory as well as multi-memory QCCs
will be constructed, although we focus the attention on the construction of unitmemory codes.
Let us recall the following two useful lemmas.
173
7.4 More QCCs from BCH Codes
We describe in this part how to obtain families of unit-memory as well as multimemory convolutional stabilizer codes with good parameters. The content presented
here can be found in the paper [93]. The technique of construction employed in the
sequence is similar as the utilized in Sect. 7.3. Our constructions differ from those
shown given in [6] at least in two aspects: (1) we construct unit-memory and also
multi-memory QCCs whereas in [6] only unit-memory QCCs were constructed; (2)
we make use directly of minimal polynomials to define classical BCH codes utilized
in our quantum code constructions.
The first construction generates quantum convolutional codes of length n = q
4
−
1, (q ≥ 3 is a prime power), with parameters
• [(n, n − 4(i − 2) − 2, 1; 2, d f ≥ i + 1)] q , 3 ≤ i ≤ q
2
− 1;
• [(n, n − 4i − 2, 1; 2 j, d f ≥ i + j + 2)] q , 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2.
In the second, we show how to obtain quantum convolutional codes of length
n = q
2m
− 1, where q ≥ 4 is a prime power and m = or d n (q
2
) ≥ 3, derived from
the Hermitian construction. Our codes have parameters:
• [(n, n − 2m(2q
2
− 3) − 2, 1; m, d f ≥ 2q
2
+ 2)] q ;
• [(n, n − 2mi − 2, 1; m j, d f ≥ i + j + 2)] q , where 1 ≤ i = j ≤ q
2
− 2;
• [(n, n − 2m(i − 1) − 2, 1; m, d f ≥ i + 2)] q , where 1 ≤ i < q
2
− 1;
• [(n, n − 2m(q
2
− 2) − 2, 1; m, d f ≥ q
2
+ 2)] q ;
• [(n, n − 2m(i + q
2
− 2) − 2, 1; m, d f ≥ i + q
2
+ 2)] q , where 1 ≤ i < q
2
− 1;
• [(n, n − 2m(i − 2) − 2, 2; 2m, d f ≥ i + 2)] q , where 3 ≤ i < q
2
− 1;
• [(n, n − 2m(i − μ) − 2, μ; mμ, d f ≥ i − μ + 4)] q , where μ ≥ 3 and μ + 1 ≤
i < q
2
− 1.
Finally, the third construction proposed provides convolutional stabilizer codes of
length n = q
m
− 1, (q ≥ 4 is a prime power and m = or d n (q) ≥ 3), derived from
the Euclidean construction, with parameters
• [(n, n − 2m(c − 1) − 2, 1; m, d f ≥ c + 2)] q , where 2 ≤ c = i + j ≤ q − 2 and
i, j ≥ 1;
• [(n, n − 2mi − 2, 1; m j, d f ≥ i + j + 2)] q , where 1 ≤ i = j ≤ q − 2;
• [(n, n − 2m(q − 2) − 2, 1; m, d f ≥ q + 2)] q ;
• [(n, n − 2m(q − 1), 1; m + 1, d f ≥ q + 3)] q ;
• [(n, n − 2m(q − 1) − 2, 1; m j, d f ≥ q + j + 2)] q , where 1 ≤ j < q − 1;
• [(n, n − 2m(2q − 3), 1; m, d f ≥ 2q + 1)] q .
As we can see above, as families of unit-memory as well as multi-memory QCCs
will be constructed, although we focus the attention on the construction of unitmemory codes.
Let us recall the following two useful lemmas.