204
7 Constructions of QCCs
and
G 2 (D) =
1 α
(a−i+2)
α
2(a−i+2)
· · · α
(n−1)(a−i+2)
+
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
D.
The code V 1 , generated by G 1 (D), is a unit-memory code of dimension k 1 = 2(i −
1) and degree γ 1 = 4; V 1 is an (n, 2[i − 1], 4; 1, [d 1 ] f ≥ n − 2i − 1) q code. Its
Euclidean dual code V
⊥
1 has parameters (n, n − 2[i − 1], 4; μ
⊥
1 , [d 1 ]
⊥
f ≥ 2i + 2) q .
The code V 2 , generated by G 2 (D), is an (n, 2, 2; 1, [d 2 ] f ) q code, so V
⊥
2 has parameters (n, n − 2, 2; μ
⊥
2 , [d 2 ]
⊥
f ≥ 3) q . From construction, it follows that V 2 ⊂ V 1 , so
V
⊥
1 ⊂ V
⊥
2 . Consider the stabilizer matrix given by
H 1 (D) | 0
0 | G 2 (D)
,
where H 1 (D) is a parity check matrix of the code V
⊥
1 . The corresponding CSS-type
code has K = 2i − 4, γ = 6, (d z ) f ≥ n − 2i − 1 and (d x ) f ≥ 3. Thus there exists
an [(n, 2i − 4, μ
∗
; 6, [d z ] f /[d x ] f )] q AQCC.
Remark 7.7.1 It is interesting to note that the idea of construction of the matrix
G 2 (D) shown in the proof of Theorem 7.7.3 is distinct from that given in Theorem 7.7.1.
Theorem 7.7.4 Let q = 2
l , where l ≥ 4 and consider that n = q + 1 and a =
q
2
.
Then there exist AQCCs with parameters
(a) [(n, 2i − 2t − 2, μ
∗
; 6, [d z ] f /[d x ] f )] q , where (d z ) f ≥ n − 2i − 1, (d x ) f ≥
2t + 3, i and t are positive integers such that 1 ≤ t ≤ i − 2 and 3 ≤ i ≤ a − 1;
(b) [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q , where (d z ) f ≥ n − 2i − 1, (d x ) f ≥ 2t + 3,
i and t are positive integers such that 1 ≤ t ≤ i − 1 and 2 ≤ i ≤ a − 1.
Proof We only show Item (a) since Item (b) is similar. The notation and the matrix
H is the same as in the proof of Theorem 7.7.3. We split H into disjoint submatrices
in order to construct a reduced basic generator matrix G 1 (D) of V 1 given by
G 1 (D) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 α
[a−(t+1)]
α
2[a−(t+1)]
· · · α
(n−1)[a−(t+1)]
1
α
a
· · ·
· · ·
α
(n−1)a
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
. . .
. . .
. . .
. . .
. . .
1 α
[a−(t−1)]
α
2[a−(t−1)]
· · · α
(n−1)[a−(t−1)]
−
−
−
−
−
1 α
[a−(t+2)]
α
2[a−(t+2)]
· · · α
(n−1)[a−(t+2)]
. . .
. . .
. . .
. . .
. . .
1 α
[a−(i−2)]
α
2[a−(i−2)]
· · · α
(n−1)[a−(i−2)]
1 α
[a−(i−1)]
α
2[a−(i−1)]
· · · α
(n−1)[a−(i−1)]
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
7 Constructions of QCCs
and
G 2 (D) =
1 α
(a−i+2)
α
2(a−i+2)
· · · α
(n−1)(a−i+2)
+
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
D.
The code V 1 , generated by G 1 (D), is a unit-memory code of dimension k 1 = 2(i −
1) and degree γ 1 = 4; V 1 is an (n, 2[i − 1], 4; 1, [d 1 ] f ≥ n − 2i − 1) q code. Its
Euclidean dual code V
⊥
1 has parameters (n, n − 2[i − 1], 4; μ
⊥
1 , [d 1 ]
⊥
f ≥ 2i + 2) q .
The code V 2 , generated by G 2 (D), is an (n, 2, 2; 1, [d 2 ] f ) q code, so V
⊥
2 has parameters (n, n − 2, 2; μ
⊥
2 , [d 2 ]
⊥
f ≥ 3) q . From construction, it follows that V 2 ⊂ V 1 , so
V
⊥
1 ⊂ V
⊥
2 . Consider the stabilizer matrix given by
H 1 (D) | 0
0 | G 2 (D)
,
where H 1 (D) is a parity check matrix of the code V
⊥
1 . The corresponding CSS-type
code has K = 2i − 4, γ = 6, (d z ) f ≥ n − 2i − 1 and (d x ) f ≥ 3. Thus there exists
an [(n, 2i − 4, μ
∗
; 6, [d z ] f /[d x ] f )] q AQCC.
Remark 7.7.1 It is interesting to note that the idea of construction of the matrix
G 2 (D) shown in the proof of Theorem 7.7.3 is distinct from that given in Theorem 7.7.1.
Theorem 7.7.4 Let q = 2
l , where l ≥ 4 and consider that n = q + 1 and a =
q
2
.
Then there exist AQCCs with parameters
(a) [(n, 2i − 2t − 2, μ
∗
; 6, [d z ] f /[d x ] f )] q , where (d z ) f ≥ n − 2i − 1, (d x ) f ≥
2t + 3, i and t are positive integers such that 1 ≤ t ≤ i − 2 and 3 ≤ i ≤ a − 1;
(b) [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q , where (d z ) f ≥ n − 2i − 1, (d x ) f ≥ 2t + 3,
i and t are positive integers such that 1 ≤ t ≤ i − 1 and 2 ≤ i ≤ a − 1.
Proof We only show Item (a) since Item (b) is similar. The notation and the matrix
H is the same as in the proof of Theorem 7.7.3. We split H into disjoint submatrices
in order to construct a reduced basic generator matrix G 1 (D) of V 1 given by
G 1 (D) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 α
[a−(t+1)]
α
2[a−(t+1)]
· · · α
(n−1)[a−(t+1)]
1
α
a
· · ·
· · ·
α
(n−1)a
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
. . .
. . .
. . .
. . .
. . .
1 α
[a−(t−1)]
α
2[a−(t−1)]
· · · α
(n−1)[a−(t−1)]
−
−
−
−
−
1 α
[a−(t+2)]
α
2[a−(t+2)]
· · · α
(n−1)[a−(t+2)]
. . .
. . .
. . .
. . .
. . .
1 α
[a−(i−2)]
α
2[a−(i−2)]
· · · α
(n−1)[a−(i−2)]
1 α
[a−(i−1)]
α
2[a−(i−1)]
· · · α
(n−1)[a−(i−1)]
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+