176
7 Constructions of QCCs
H 2,q 2 +i =
1 α
(q
2 +i)
α
2(q
2 +i)
· · · α
(n−1)(q
2 +i)
with respect to β. The new matrix is denoted by H 1 and since C 1 has dimension
n − 2, H 1 has 2 linearly independent rows.
We know that rk H 0 ≥ rk H 1 . Consider the convolutional code V generated by
G(D) = ˜
H 0 + ˜
H 1 D,
where ˜
H 0 = H 0 and ˜
H 1 is obtained from H 1 by adding zero-rows at the bottom such
that ˜
H 1 has the number of rows of H 0 in total. By construction, V has dimension
2(i − 2) + 1 and degree δ V = 2; hence, V is an (n, 2(i − 2) + 1, 2; 1, d f ∗ ) q 2 code.
The Euclidean dual V
⊥ of V has dimension n − 2(i − 2) − 1 and degree 2. Let us
now compute the free distance d
⊥
f of V
⊥ . By Theorem 7.1.1 Item (b), the free distance
of V
⊥ is bounded by min{d 0 + d 1 , d} ≤ d
⊥
f ≤ d, where d i is the minimum distance of
the code C i = {v ∈ F
n
q | v ˜
H
t
i = 0}. From construction one has d ≥ i + 1, d 0 ≥ i and
d 1 ≥ 2, so d
⊥
f ≥ i + 1 and V
⊥ has parameters (n, n − 2(i − 2) − 1, 2; μ, d
⊥
f ≥ i +
1) q 2 for each 3 ≤ i ≤ q
2
− 1. The codes V
⊥ and V
⊥ h have the same degree as code
(see the proof of Theorem 7 in [7]). Since wt(V
⊥
) = wt(V
⊥ h ), the convolutional code
V
⊥ h has parameters (n, n − 2(i − 2) − 1, 2; m
∗
, d
⊥ h
f ≥ i + 1) q 2 . From Lemma 7.4.4
and from Theorem 7.1.1 Item (c), we have V ⊂ V
⊥ h . Applying Lemma 7.3.1, there
exists an [(n, n − 4(i − 2) − 2, 1; 2, d f ≥ i + 1)] q convolutional stabilizer code, for
each 3 ≤ i ≤ q
2
− 1.
The next theorem generates more new QCCs:
Theorem 7.4.2 Let n = q
4
− 1 where q ≥ 3 is a prime power. Then there exists an
[(n, n − 4i − 2, 1; 2 j, d f ≥ i + j+ 2)] q QCC, where 1 ≤ i = j and 2 ≤ i + j ≤
q
2
− 2.
Proof Let C be the BCH code generated by
M
(q
2 +1)
(x)M
(q
2 +2)
(x) · · · · · · · · M
(q
2 +i+ j)
(x)M
(q
2 +i+ j+1)
(x),
where 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2. C has a parity check matrix H . Suppose
that C 0 is the BCH code generated by
M
(q
2 +1)
. . . M
(q
2 +i)
(x)M
(q
2 +i+1)
(x),
where 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2; C 0 has parity check matrix H 0 . Assume
also that C 1 is the BCH code generated by
M
(q
2 +i+2)
. . . M
(q
2 +i+ j+1)
(x),
where 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2; C 1 has parity check matrix H 1 . Applying
Lemma 7.4.3 one can easily verify that C has parameters [n, n − 2(i + j) − 1, d ≥
7 Constructions of QCCs
H 2,q 2 +i =
1 α
(q
2 +i)
α
2(q
2 +i)
· · · α
(n−1)(q
2 +i)
with respect to β. The new matrix is denoted by H 1 and since C 1 has dimension
n − 2, H 1 has 2 linearly independent rows.
We know that rk H 0 ≥ rk H 1 . Consider the convolutional code V generated by
G(D) = ˜
H 0 + ˜
H 1 D,
where ˜
H 0 = H 0 and ˜
H 1 is obtained from H 1 by adding zero-rows at the bottom such
that ˜
H 1 has the number of rows of H 0 in total. By construction, V has dimension
2(i − 2) + 1 and degree δ V = 2; hence, V is an (n, 2(i − 2) + 1, 2; 1, d f ∗ ) q 2 code.
The Euclidean dual V
⊥ of V has dimension n − 2(i − 2) − 1 and degree 2. Let us
now compute the free distance d
⊥
f of V
⊥ . By Theorem 7.1.1 Item (b), the free distance
of V
⊥ is bounded by min{d 0 + d 1 , d} ≤ d
⊥
f ≤ d, where d i is the minimum distance of
the code C i = {v ∈ F
n
q | v ˜
H
t
i = 0}. From construction one has d ≥ i + 1, d 0 ≥ i and
d 1 ≥ 2, so d
⊥
f ≥ i + 1 and V
⊥ has parameters (n, n − 2(i − 2) − 1, 2; μ, d
⊥
f ≥ i +
1) q 2 for each 3 ≤ i ≤ q
2
− 1. The codes V
⊥ and V
⊥ h have the same degree as code
(see the proof of Theorem 7 in [7]). Since wt(V
⊥
) = wt(V
⊥ h ), the convolutional code
V
⊥ h has parameters (n, n − 2(i − 2) − 1, 2; m
∗
, d
⊥ h
f ≥ i + 1) q 2 . From Lemma 7.4.4
and from Theorem 7.1.1 Item (c), we have V ⊂ V
⊥ h . Applying Lemma 7.3.1, there
exists an [(n, n − 4(i − 2) − 2, 1; 2, d f ≥ i + 1)] q convolutional stabilizer code, for
each 3 ≤ i ≤ q
2
− 1.
The next theorem generates more new QCCs:
Theorem 7.4.2 Let n = q
4
− 1 where q ≥ 3 is a prime power. Then there exists an
[(n, n − 4i − 2, 1; 2 j, d f ≥ i + j+ 2)] q QCC, where 1 ≤ i = j and 2 ≤ i + j ≤
q
2
− 2.
Proof Let C be the BCH code generated by
M
(q
2 +1)
(x)M
(q
2 +2)
(x) · · · · · · · · M
(q
2 +i+ j)
(x)M
(q
2 +i+ j+1)
(x),
where 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2. C has a parity check matrix H . Suppose
that C 0 is the BCH code generated by
M
(q
2 +1)
. . . M
(q
2 +i)
(x)M
(q
2 +i+1)
(x),
where 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2; C 0 has parity check matrix H 0 . Assume
also that C 1 is the BCH code generated by
M
(q
2 +i+2)
. . . M
(q
2 +i+ j+1)
(x),
where 1 ≤ i = j and 2 ≤ i + j ≤ q
2
− 2; C 1 has parity check matrix H 1 . Applying
Lemma 7.4.3 one can easily verify that C has parameters [n, n − 2(i + j) − 1, d ≥