174
7 Constructions of QCCs
Lemma 7.4.1 ([6, Proposition 1]) Let C be an (n, (n − k)/2, γ; μ) q convolutional
code such that C ⊂ C
⊥ . Then there exists an [(n, k, μ; γ, d f )] q convolutional stabilizer code, where d f = wt (C
⊥
\C).
Lemma 7.4.2 ([6, Proposition 2]) Let C be an (n, (n − k)/2, γ; μ) q 2 convolutional
code such that C ⊂ C
⊥ h . Then there exists an [(n, k, μ; γ, d f )] q convolutional stabilizer code, where d f = wt (C
⊥ h \C).
7.4.1 Construction I
Here we focus on the construction of convolutional stabilizer codes of length q
4
− 1
over F q 2 . To proceed further we need some results available in [89].
Lemma 7.4.3 Let q ≥ 3 be a prime power and n = q
4
− 1. Consider the q
2
− 1
q
2 -ary cosets modulo n given by
C [q 2 +1] ,
C [q 2 +2] = {q
2
+ 2, 1 + 2q
2
},
. . .
C [2q 2 −1] = {2q
2
− 1, 1 + (q
2
− 1)q
2
}.
Then the following hold:
(a) the q
2 -ary coset C [q 2 +1] contains only one element;
(b) each of the other cosets contains two elements;
(c) all these q
2 -cosets are mutually disjoints.
Proof The proof can be found in [89, Lemma 3.2].
Lemma 7.4.4 Let n = q
4
− 1 where q ≥ 3 is a prime power. Let C be the cyclic
code of length n over F q 2 generated by the product of the minimal polynomials
M
(q
2 +1)
(x)M
(q
2 +2)
(x) . . . M
(q
2 + j)
(x),
1 ≤ j ≤ q
2
− 1. Then C is Hermitian self-orthogonal.
Proof See [89, Theorem III.1].
At this point we are ready to show Theorem 7.4.1.
Theorem 7.4.1 Let n = q
4
− 1, where q ≥ 3 is a prime power. Then there exists an
[(n, n − 4(i − 2) − 2, 1; 2, d f ≥ i +1)] q quantum convolutional code, where 3 ≤
i ≤ q
2
− 1.