7.4 More QCCs from BCH Codes
181
where 3 ≤ i < q
2
− 1. We know from Lemma 7.4.5 that C has parameters [n, n −
mi − 1, d ≥ i + 2] q 2 , where 3 ≤ i < q
2
− 1. A parity check matrix of C is the matrix
H .
Let C 0 be the BCH code generated by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i−2)
(x),
where 3 ≤ i < q
2
− 1. From Lemma 7.4.5, C 0 has parameters [n, n − m(i − 2) − 1,
d 0 ≥ i] q 2 , 3 ≤ i < q
2
− 1. A parity check matrix of C 0 is H 0 .
Assume also that C 1 is the BCH code generated by M
(s+i−1)
(x), and let C 2 be
the BCH code generated by M
(s+i)
(x), where 3 ≤ i < q
2
− 1. We know that C 1 and
C 2 have parameters [n, n − m, d 1 ≥ 2] q 2 and [n, n − m, d 2 ≥ 2] q 2 , respectively. A
parity check matrices of C 1 and C 2 are, respectively, H 1 and H 2 .
The convolutional code V generated by the matrix G(D) = ˜
H 0 + ˜
H 1 D + ˜
H 2 D
2
has parameters (n, m(i − 2) + 1, 2m; 2, d f ∗ ) q 2 . Note that γ = 2m because, from
Lemma 7.4.5, since the q
2 -ary coset C [s+i] contains m elements, it follows that H 2
have m linearly independent rows, and each of the first m linearly independent rows
of ˜
H 2 has degree 2.
We know that rk H 0 ≥ rk H 1 and rk H 0 ≥ rk H 2 . The code V
⊥ h is an (n, n −
m(i − 2) − 1, 2m; μ, d
⊥ h
f ≥ i + 2) q 2 code, where we compute the free distance
d
⊥ h
f
by applying Theorem 7.1.1 (Item (b)). From Lemma 7.4.6 and by Theorem 7.1.1, Item (b), one has V ⊂ V
⊥ h . Applying Lemma 7.3.1, there exists an
[(n, n − 2m(i − 2) − 2, 2; 2m, d f ≥ i + 2)] q QCC, for each 3 ≤ i < q
2
− 1.
Theorem 7.4.5 can be generalized as follows.
Theorem 7.4.6 (multi-memory QCCs) Let n = q
2m
− 1, where q ≥ 4 is a prime
power and m = ord n (q
2
) ≥ 3. Then there exists an
[(n, n − 2m(i − μ) − 2, μ; mμ, d f ≥ i − μ + 4)] q
QCC, where μ ≥ 3 and μ + 1 ≤ i < q
2
− 1.
Proof Let C be the BCH code generated by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i)
(x),
where μ ≥ 3 and μ + 1 ≤ i < q
2
− 1. Assume that C 0 is the BCH code generated
by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i−μ)
(x),
where μ + 1 ≤ i < q
2
− 1. Assume also that C j , for j = 1, . . . , μ, is the BCH code
generated by
M
(s+i−μ+ j)
(x),
181
where 3 ≤ i < q
2
− 1. We know from Lemma 7.4.5 that C has parameters [n, n −
mi − 1, d ≥ i + 2] q 2 , where 3 ≤ i < q
2
− 1. A parity check matrix of C is the matrix
H .
Let C 0 be the BCH code generated by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i−2)
(x),
where 3 ≤ i < q
2
− 1. From Lemma 7.4.5, C 0 has parameters [n, n − m(i − 2) − 1,
d 0 ≥ i] q 2 , 3 ≤ i < q
2
− 1. A parity check matrix of C 0 is H 0 .
Assume also that C 1 is the BCH code generated by M
(s+i−1)
(x), and let C 2 be
the BCH code generated by M
(s+i)
(x), where 3 ≤ i < q
2
− 1. We know that C 1 and
C 2 have parameters [n, n − m, d 1 ≥ 2] q 2 and [n, n − m, d 2 ≥ 2] q 2 , respectively. A
parity check matrices of C 1 and C 2 are, respectively, H 1 and H 2 .
The convolutional code V generated by the matrix G(D) = ˜
H 0 + ˜
H 1 D + ˜
H 2 D
2
has parameters (n, m(i − 2) + 1, 2m; 2, d f ∗ ) q 2 . Note that γ = 2m because, from
Lemma 7.4.5, since the q
2 -ary coset C [s+i] contains m elements, it follows that H 2
have m linearly independent rows, and each of the first m linearly independent rows
of ˜
H 2 has degree 2.
We know that rk H 0 ≥ rk H 1 and rk H 0 ≥ rk H 2 . The code V
⊥ h is an (n, n −
m(i − 2) − 1, 2m; μ, d
⊥ h
f ≥ i + 2) q 2 code, where we compute the free distance
d
⊥ h
f
by applying Theorem 7.1.1 (Item (b)). From Lemma 7.4.6 and by Theorem 7.1.1, Item (b), one has V ⊂ V
⊥ h . Applying Lemma 7.3.1, there exists an
[(n, n − 2m(i − 2) − 2, 2; 2m, d f ≥ i + 2)] q QCC, for each 3 ≤ i < q
2
− 1.
Theorem 7.4.5 can be generalized as follows.
Theorem 7.4.6 (multi-memory QCCs) Let n = q
2m
− 1, where q ≥ 4 is a prime
power and m = ord n (q
2
) ≥ 3. Then there exists an
[(n, n − 2m(i − μ) − 2, μ; mμ, d f ≥ i − μ + 4)] q
QCC, where μ ≥ 3 and μ + 1 ≤ i < q
2
− 1.
Proof Let C be the BCH code generated by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i)
(x),
where μ ≥ 3 and μ + 1 ≤ i < q
2
− 1. Assume that C 0 is the BCH code generated
by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i−μ)
(x),
where μ + 1 ≤ i < q
2
− 1. Assume also that C j , for j = 1, . . . , μ, is the BCH code
generated by
M
(s+i−μ+ j)
(x),