Chapter 7
Constructions of QCCs
This chapter is devoted to the construction of quantum convolutional codes with good
or even optimal parameters (in the sense that the code parameters attain the generalized quantum Singleton bound). We begin by presenting the necessary background
for the constructions, i.e., the class of classical convolutional codes.
In Sects. 7.1 and 7.2, we give the necessary background to our code constructions. Families of maximum-distance-separable (optimal) quantum convolutional
BCH codes are presented in Sect. 7.3. In Sect. 7.4, we construct more families of
quantum convolutional BCH codes. In Sect. 7.5, we construct families of quantum
convolutional codes derived from negacyclic codes. More families of quantum convolutional codes derived from algebraic geometry are presented in Sect. 7.6. Finally,
in Sect. 7.7, we introduce the first class of asymmetric quantum convolutional codes
displayed in the literature and we show how to construct such codes.
7.1 Convolutional Codes
The class of classical convolutional codes is extensively investigated in the literature
[29, 40, 75, 107, 130, 135, 145]. Forney [40] was one of the pioneers in the investigations concerning convolutional codes. Constructions of convolutional codes with
good parameters or even maximum-distance-separable (MDS) convolutional codes
(optimal in the sense that they attain the generalized Singleton bound [135]) have
also been presented in the literature [40, 51, 98, 99, 101, 107, 130, 135, 145].
Rosenthal et al. introduced the generalized Singleton bound for convolutional codes
in 1999 [135] (see also [145]).
A convolutional code of length n and rank k can be understood as a free module
(see Definition 1.5.4) over some finite field F q given by a direct summand F q [D]
n .
Since free modules have basis, we can codify it by means of a polynomial matrix
with suitable properties.
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_7
163
Constructions of QCCs
This chapter is devoted to the construction of quantum convolutional codes with good
or even optimal parameters (in the sense that the code parameters attain the generalized quantum Singleton bound). We begin by presenting the necessary background
for the constructions, i.e., the class of classical convolutional codes.
In Sects. 7.1 and 7.2, we give the necessary background to our code constructions. Families of maximum-distance-separable (optimal) quantum convolutional
BCH codes are presented in Sect. 7.3. In Sect. 7.4, we construct more families of
quantum convolutional BCH codes. In Sect. 7.5, we construct families of quantum
convolutional codes derived from negacyclic codes. More families of quantum convolutional codes derived from algebraic geometry are presented in Sect. 7.6. Finally,
in Sect. 7.7, we introduce the first class of asymmetric quantum convolutional codes
displayed in the literature and we show how to construct such codes.
7.1 Convolutional Codes
The class of classical convolutional codes is extensively investigated in the literature
[29, 40, 75, 107, 130, 135, 145]. Forney [40] was one of the pioneers in the investigations concerning convolutional codes. Constructions of convolutional codes with
good parameters or even maximum-distance-separable (MDS) convolutional codes
(optimal in the sense that they attain the generalized Singleton bound [135]) have
also been presented in the literature [40, 51, 98, 99, 101, 107, 130, 135, 145].
Rosenthal et al. introduced the generalized Singleton bound for convolutional codes
in 1999 [135] (see also [145]).
A convolutional code of length n and rank k can be understood as a free module
(see Definition 1.5.4) over some finite field F q given by a direct summand F q [D]
n .
Since free modules have basis, we can codify it by means of a polynomial matrix
with suitable properties.
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_7
163