7.4 More QCCs from BCH Codes
177
i + j + 2] q 2 , C 0 has parameters [n, n − 2i − 1, d 0 ≥ i + 2] q 2 and C 1 has parameters
[n, n − 2 j, d 1 ≥ j + 1] q 2 .
The convolutional code V generated by the reduced basic generator matrix
G(D) = ˜
H 0 + ˜
H 1 D, is a unit-memory convolutional code of dimension 2i + 1 and
degree δ V = 2 j, so V has parameters (n, 2i + 1, 2 j; 1, d f ∗ ) q 2 . The convolutional
code V
⊥ h has parameters (n, n − 2i − 1, 2 j; μ, d
⊥ h
f ≥ i + j + 2) q 2 , where the lower
bound for the free distance i + j + 2 was found by applying Theorem 7.1.1 Item (b).
From Lemma 7.4.4 and from Theorem 7.1.1 Item (b), one has V ⊂ V
⊥ h . Applying
Lemma 7.3.1, there exists an [(n, n − 4i − 2, 1; 2 j, d f ≥ i + j + 2)] q QCC, for
each 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2.
Example 7.4.1 In Theorem 7.3.2 consider that q = 5 and i = 8. Let C be the BCH
code of length 624 over F 25 , generated by
C = =M
(26)
(x)M
(27)
(x) . . . M
(32)
(x)M
(33)
(x),
C 0 be the BCH code of length 624 over F 25 , generated by
C 0 = =M
(26)
(x)M
(27)
(x) . . . M
(31)
(x)M
(32)
(x),
and suppose also that C 1 is the BCH code of length 624 over F 25 , generated by
M
(33)
(x). Applying Theorem 7.4.1 one has an [(624, 598, 1; 2, d f ≥ 9)] 5 quantum
convolutional code.
Analogously, in Theorem 7.4.2, let us consider that q = 5 and i = j = 3. Let C
be the BCH code generated by
M
(26)
(x)M
(27)
(x) . . . M
(32)
(x),
C 0 be the BCH code generated by
M
(26) M
(27)
(x)M
(28)
(x)M
(29)
(x).
Suppose that C 1 is the BCH code generated by
M
(30)
. . . M
(32)
(x).
Applying Theorem 7.4.2, we obtain an [(624, 610, 1; 6, d f ≥ 8)] 5 QCC.
7.4.2 Construction II
In this subsection, we apply similar technique which was developed in the previous
subsection in order to obtain more convolutional stabilizer codes. Lemmas 7.4.5 and
7.4.6 are essentials to our constructions.