206
7 Constructions of QCCs
Example 7.7.1 We now list the parameters of some AQCCs obtaining by applying
Theorem 7.7.4: [(17, 6, μ
∗
; 6, [d z ] f ≥ 6/[d x ] f ≥ 5)] 16 ,
[(17, 8, μ
∗
; 4, [d z ] f ≥ 6/[d x ] f ≥ 5)] 16 ,
[(17, 8, μ
∗
; 6, [d z ] f ≥ 5/[d x ] f ≥ 4)] 16 ,
[(17, 10, μ
∗
; 4, [d z ] f ≥ 5/[d x ] f ≥ 4)] 16 , [(33, 24, μ
∗
; 6, [d z ] f ≥ 5/[d x ] f ≥ 4)] 32 ,
[(33, 26, μ
∗
; 4, [d z ] f ≥ 5/[d x ] f ≥ 4)] 32 , [(33, 22, μ
∗
; 6, [d z ] f ≥ 6/[d x ] f ≥ 5)] 32 ,
[(33, 24, μ
∗
; 4, [d z ] f ≥ 6/[d x ] f ≥ 5)] 32 , [(33, 20, μ
∗
; 6, [d z ] f ≥ 8/[d x ] f ≥ 5)] 32 ,
[(33, 22, μ
∗
; 4, [d z ] f ≥ 8/[d x ] f ≥ 5)] 32 .
More families of AQCCs can be obtained by applying Theorem 7.7.5.
Theorem 7.7.5 Assume that q = p
l , where p is an odd prime and l ≥ 2. Consider
that n = q + 1 and a =
n
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 and (d x ) f ≥ 2t +
2, for all 1 ≤ t ≤ i − 2, where 3 ≤ i ≤ a − 1;
(b) [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q , where (d z ) f ≥ n − 2i and (d x ) f ≥ 2t + 2,
for all 1 ≤ t ≤ i − 1, where 2 ≤ i ≤ a − 1.
Proof Analogous to that of Theorem 7.7.4.
Exercise 7.7.1 Show Theorem 7.7.5.
Remark 7.7.2 We call the attention that the idea of construction of the matrix G 2 (D)
is different for each of Theorems 7.7.1, 7.7.3, 7.7.4 and 7.7.5.
7.7.3 Construction III
We here are interested in the construction of AQCCs derived from Reed–Solomon
(RS) and generalized Reed–Solomon (GRS) codes. We first deal with RS codes.
Recall that a RS code over F q is a BCH code, of length n = q − 1, with parameters
[n, n − d + 1, d] q , where 2 ≤ d ≤ n. A parity check matrix of a RS code is given
by
H δ,b =
⎡
⎢
⎢
⎢
⎣
1 α
b
α
2b
· · · α
(n−1)b
1 α
(b+1)
α
2(b+1)
· · · α
(n−1)(b+1)
. . .
. . .
. . .
. . .
. . .
1 α
(b+d−2)
· · · · · · α
(n−1)(b+d−2)
⎤
⎥
⎥
⎥
⎦
.
In Theorem 7.7.6 presented in the sequence, we construct AQCCs derived from
RS codes:
Theorem 7.7.6 Assume that q ≥ 8 is a prime power. Then there exist AQCCs with
parameters
(a) [(q − 1, i − t − 1, μ
∗
; 3, [d z ] f /[d x ] f )] q , where (d z ) f ≥ q − i − 1, (d x ) f ≥ t +
2, for all 1 ≤ t ≤ i − 2, where 3 ≤ i ≤ q − 3;
7 Constructions of QCCs
Example 7.7.1 We now list the parameters of some AQCCs obtaining by applying
Theorem 7.7.4: [(17, 6, μ
∗
; 6, [d z ] f ≥ 6/[d x ] f ≥ 5)] 16 ,
[(17, 8, μ
∗
; 4, [d z ] f ≥ 6/[d x ] f ≥ 5)] 16 ,
[(17, 8, μ
∗
; 6, [d z ] f ≥ 5/[d x ] f ≥ 4)] 16 ,
[(17, 10, μ
∗
; 4, [d z ] f ≥ 5/[d x ] f ≥ 4)] 16 , [(33, 24, μ
∗
; 6, [d z ] f ≥ 5/[d x ] f ≥ 4)] 32 ,
[(33, 26, μ
∗
; 4, [d z ] f ≥ 5/[d x ] f ≥ 4)] 32 , [(33, 22, μ
∗
; 6, [d z ] f ≥ 6/[d x ] f ≥ 5)] 32 ,
[(33, 24, μ
∗
; 4, [d z ] f ≥ 6/[d x ] f ≥ 5)] 32 , [(33, 20, μ
∗
; 6, [d z ] f ≥ 8/[d x ] f ≥ 5)] 32 ,
[(33, 22, μ
∗
; 4, [d z ] f ≥ 8/[d x ] f ≥ 5)] 32 .
More families of AQCCs can be obtained by applying Theorem 7.7.5.
Theorem 7.7.5 Assume that q = p
l , where p is an odd prime and l ≥ 2. Consider
that n = q + 1 and a =
n
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 and (d x ) f ≥ 2t +
2, for all 1 ≤ t ≤ i − 2, where 3 ≤ i ≤ a − 1;
(b) [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q , where (d z ) f ≥ n − 2i and (d x ) f ≥ 2t + 2,
for all 1 ≤ t ≤ i − 1, where 2 ≤ i ≤ a − 1.
Proof Analogous to that of Theorem 7.7.4.
Exercise 7.7.1 Show Theorem 7.7.5.
Remark 7.7.2 We call the attention that the idea of construction of the matrix G 2 (D)
is different for each of Theorems 7.7.1, 7.7.3, 7.7.4 and 7.7.5.
7.7.3 Construction III
We here are interested in the construction of AQCCs derived from Reed–Solomon
(RS) and generalized Reed–Solomon (GRS) codes. We first deal with RS codes.
Recall that a RS code over F q is a BCH code, of length n = q − 1, with parameters
[n, n − d + 1, d] q , where 2 ≤ d ≤ n. A parity check matrix of a RS code is given
by
H δ,b =
⎡
⎢
⎢
⎢
⎣
1 α
b
α
2b
· · · α
(n−1)b
1 α
(b+1)
α
2(b+1)
· · · α
(n−1)(b+1)
. . .
. . .
. . .
. . .
. . .
1 α
(b+d−2)
· · · · · · α
(n−1)(b+d−2)
⎤
⎥
⎥
⎥
⎦
.
In Theorem 7.7.6 presented in the sequence, we construct AQCCs derived from
RS codes:
Theorem 7.7.6 Assume that q ≥ 8 is a prime power. Then there exist AQCCs with
parameters
(a) [(q − 1, i − t − 1, μ
∗
; 3, [d z ] f /[d x ] f )] q , where (d z ) f ≥ q − i − 1, (d x ) f ≥ t +
2, for all 1 ≤ t ≤ i − 2, where 3 ≤ i ≤ q − 3;