7.7 AQCCs
203
where each entry is replaced by the corresponding column of l elements from F q ,
where l = ord n (q), and then removing any linearly dependent rows. The rows of the
resulting matrix over F q are the parity checks satisfied by C.
We need to utilize useful results shown in [98].
Theorem 7.7.2 ([98, Theorem 4.2]) Assume that q = 2
t , where t ≥ 3 is an integer,
n = q + 1 and consider that a =
q
2
. Then there exists an (n, n − 2i, 2; 1, 2i + 3) q ,
classical MDS convolutional code, where 1 ≤ i ≤ a − 1.
Theorem 7.7.3 establishes conditions to construct AQCCs derived from BCH
codes.
Theorem 7.7.3 Let q = 2
t , where t ≥ 4 and consider that n = q + 1 and a =
q
2
.
Then there exists an [(n, 2i − 4, μ
∗
; 6, [d z ] f /[d x ] f )] q AQCC, where (d z ) f ≥ n −
2i − 1 and (d x ) f ≥ 3, for all 3 ≤ i ≤ a − 1.
Proof Consider the parity check F q -matrix of the BCH code C given by
H =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
a
· · ·
· · · α
(n−1)a
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
. . .
. . .
. . .
. . .
. . .
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
whose entries are expanded with respect to some F q -basis B of F q 2 , after removing
the linearly dependent rows. This BCH code was constructed in the proof of [98,
Theorem 4.2] (more precisely, it is the code C 2 constructed there); C is a MDS
code with parameters [n, n − 2i − 2, 2i + 3] q . Its (Euclidean) dual code C
⊥ is also
a MDS code with parameters [n, 2i + 2, n − 2i − 1] q .
We next construct a classical convolutional code V 1 generated by the reduced
basic matrices
G 1 (D) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(a−i+2)
α
2(a−i+2)
· · · α
(n−1)(a−i+2)
−
−
−
−
−
1
α
a
· · ·
· · · α
(n−1)a
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
. . .
. . .
. . .
. . .
. . .
1 α
(a−i+3)
α
2(a−i+3)
· · · α
(n−1)(a−i+3)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
−
−
−
−
−
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
0
0
0
0
0
. . .
. . .
. . .
. . .
. . .
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
D