166
7 Constructions of QCCs
coding theory has gained strength only in the last three decades, whereas theory of
classical convolutional codes was already established in the literature a long time
ago. The precursors in the investigation of the structure of quantum convolutional
codes were Ollivier and Tillich [123, 124]. Almeida and Palazzo Jr. [32] constructed
the first concatenated [(4, 1, 3)] quantum convolutional code.
To put the reader into context, Grassl and Rötteler [58, 59] generated quantum
convolutional codes as well as they provide algorithms to obtain non-catastrophic
encoders. Forney et al., constructed rate (n − 2)/n quantum convolutional codes.
Wilde and Brun [160, 161] generated entanglement-assisted quantum convolutional
coding. Houshmand et al. [66] investigated constructions of quantum convolutional
encoders with minimal-memory. In the papers [98, 99], we constructed families
of optimal (MDS) quantum convolutional codes derived from BCH codes and from
negacyclic codes, respectively. Klappenecker et al. [5–7] constructed several families
of QCCs derived from Reed–Solomon, Reed–Muller and BCH codes. Additionally,
they derived the generalized quantum Singleton bound for quantum convolutional
codes. Zhu et al. [170] constructed optimal QCCs derived from classical constacyclic
codes.
We will briefly expose the background on QCCs. The readers who are interested
to learn more about QCCs will find a detailed description of this class of (quantum)
codes in the works [123, 124].
Definition 7.2.1 A quantum convolutional code is defined by means of its stabilizer
which is a subgroup of the infinite version of the Pauli group, consisting of tensor
products of generalized Pauli matrices acting on a semi-infinite stream of qudits. The
stabilizer can be defined by a stabilizer matrix of the form
S(D) = (X (D) | Z (D)) ∈ F q [D]
(n−k)×2n
satisfying the symplectic orthogonality condition given by
X (D)Z (1/D)
t
− Z (D)X (1/D)
t
= 0.
The constraint length and the degree of a QCC are defined similarly as in the case
of convolutional codes.
Definition 7.2.2 Let Q be a QCC defined by a full rank (as always) stabilizer matrix
S(D). The constraint length of Q is defined to be
γ i = max 1≤ j≤n {max{deg X i j (D), deg Z i j (D)}},
and the overall constraint length is defined by
γ =
n−k
i=1
γ i .
7 Constructions of QCCs
coding theory has gained strength only in the last three decades, whereas theory of
classical convolutional codes was already established in the literature a long time
ago. The precursors in the investigation of the structure of quantum convolutional
codes were Ollivier and Tillich [123, 124]. Almeida and Palazzo Jr. [32] constructed
the first concatenated [(4, 1, 3)] quantum convolutional code.
To put the reader into context, Grassl and Rötteler [58, 59] generated quantum
convolutional codes as well as they provide algorithms to obtain non-catastrophic
encoders. Forney et al., constructed rate (n − 2)/n quantum convolutional codes.
Wilde and Brun [160, 161] generated entanglement-assisted quantum convolutional
coding. Houshmand et al. [66] investigated constructions of quantum convolutional
encoders with minimal-memory. In the papers [98, 99], we constructed families
of optimal (MDS) quantum convolutional codes derived from BCH codes and from
negacyclic codes, respectively. Klappenecker et al. [5–7] constructed several families
of QCCs derived from Reed–Solomon, Reed–Muller and BCH codes. Additionally,
they derived the generalized quantum Singleton bound for quantum convolutional
codes. Zhu et al. [170] constructed optimal QCCs derived from classical constacyclic
codes.
We will briefly expose the background on QCCs. The readers who are interested
to learn more about QCCs will find a detailed description of this class of (quantum)
codes in the works [123, 124].
Definition 7.2.1 A quantum convolutional code is defined by means of its stabilizer
which is a subgroup of the infinite version of the Pauli group, consisting of tensor
products of generalized Pauli matrices acting on a semi-infinite stream of qudits. The
stabilizer can be defined by a stabilizer matrix of the form
S(D) = (X (D) | Z (D)) ∈ F q [D]
(n−k)×2n
satisfying the symplectic orthogonality condition given by
X (D)Z (1/D)
t
− Z (D)X (1/D)
t
= 0.
The constraint length and the degree of a QCC are defined similarly as in the case
of convolutional codes.
Definition 7.2.2 Let Q be a QCC defined by a full rank (as always) stabilizer matrix
S(D). The constraint length of Q is defined to be
γ i = max 1≤ j≤n {max{deg X i j (D), deg Z i j (D)}},
and the overall constraint length is defined by
γ =
n−k
i=1
γ i .