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7 Constructions of QCCs
7.3 Constructions of Optimal QCCs
In the previous subsection we defined a quantum convolutional code. Here, we construct a family of unit-memory quantum MDS convolutional codes derived from
BCH codes. The content of this part can be found in our paper [98].
The first family of QCCs that we construct is a family of MDS convolutional
codes with parameters
(n, n − 2i, 2; 1, 2i + 3) q ,
where 1 ≤ i ≤
q
2
− 1, q = 2
t , t ≥ 3 and n = q + 1. After this, we utilize these convolutional codes to construct our MDS QCCs with parameters
[(n, n − 4i, 1; 2, 2i + 3)] q ,
where 2 ≤ i ≤
q
2
− 2, q = 2
t , t ≥ 3 and n = q
2
+ 1.
It is interesting to observe that the order between the degree and the memory
is changed when comparing the parameters of classical and quantum convolutional
codes. This is usual in some papers in the literature (see for instance [6]).
The following lemma states that self-orthogonal convolutional codes can by utilized to construct quantum convolutional codes.
Lemma 7.3.1 ([6, Proposition 2]) Let C be an (n, (n − k)/2, γ; m) q 2 convolutional
code such that C ⊆ C
⊥ h . Then there exists an [(n, k, m; γ, d f )] q convolutional stabilizer code, where d f = wt(C
⊥ h \C).
Let C be an [(n, k, m; γ, d f )] q QCC. Recall that C is pure if does not exist errors
of weight less than d f in the stabilizer of C. The following result is an analogous to
the Singleton bound for quantum convolutional codes.
Theorem 7.3.1 ([7]) (Quantum Singleton bound) The free distance of an [(n, k,
m; γ, d f )] q F q 2 -linear pure convolutional stabilizer code is bounded by
d f ≤
n − k
2
2γ
n + k
+ 1
+ γ + 1.
Remark 7.3.1 It is interesting to note that this is one of the few bounds presenting in
the literature concerning quantum convolutional codes. Due to this, we recommend
for interested readers to investigate asymptotic bounds, Hamming bound and other
types of structures concerning convolutional stabilizer codes.