188
7 Constructions of QCCs
From Remark 7.5.1 and from the definition of negacyclic BCH codes, the result
follows. The proof is complete.
We are now ready to show one of the main results of this subsection.
Theorem 7.5.2 Let n = q
2
+ 1, where q ≡ 1 mod 4 is a power of an odd prime
and suppose that s = n/2. Then there exists an (n, n − 2i + 1, 2; 1, 2i + 2) q 2 MDS
convolutional code, where 2 ≤ i ≤ n/2 − 1.
Proof First, note that gcd(n, q) = 1 and or d 2n (q
2
) = 2. Let β be a primitive 2nth
root of unity in F q 2m . Consider that C 2 is the negacyclic BCH code of length n over
F q 2 generated by the product of the minimal polynomials
C 2 = =g 2 (x) = =M
(s)
(x)M
(s+2)
(x) · · · · · M
(s+2i)
(x),
where 2 ≤ i ≤ s − 1.
By Theorem 7.5.1, a parity check matrix of C 2 is obtained from the matrix
H 2 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 β
s
β
2s
· · · β
(n−1)s
1 β
(s+2)
β
2(s+2)
· · · β
(n−1)(s+2)
1 β
(s+4)
β
2(s+4)
· · · β
(n−1)(s+4)
. . .
. . .
. . .
. . .
. . .
1 β
(s+2i)
β
2(s+2i)
· · · β
(n−1)(s+2i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) with respect to some
F q 2 -basis β of F q 4 and then removing one linearly dependent row. From Lemma 7.5.1,
this new matrix H C 2 has rank 2i + 1; so, C 2 has dimension n − 2i − 1. From the
BCH bound for negacyclic codes, it follows that the minimum distance d 2 of C 2
satisfies d 2 ≥ 2i + 2. Thus, from the (classical) Singleton bound, one concludes that
C 2 is a MDS code with parameters [n, n − 2i − 1, 2i + 2] q 2 and, consequently, its
Hermitian dual code has dimension 2i + 1.
Next we assume that C 1 is the negacyclic BCH code of length n over F q 2 generated
by the product of the minimal polynomials
C 1 = =g 1 (x) = =M
(s)
(x)M
(s+2)
(x) · · · · · M
[s+2(i−1)]
(x),
Similarly, by Theorem 7.5.1, C 1 has a parity check matrix derived from the matrix
H 1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1
β
s
β
2s
· · ·
β
(n−1)s
1 β
(s+2)
β
2(s+2)
· · · β
(n−1)(s+2)
1 β
(s+4)
β
2(s+4)
· · · β
(n−1)(s+4)
. . .
. . .
. . .
. . .
. . .
1 β
[s+2(i−1)]
β
2[s+2(i−1)]
· · · β
(n−1)[s+2(i−1)]
⎤
⎥
⎥
⎥
⎥
⎥
⎦
7 Constructions of QCCs
From Remark 7.5.1 and from the definition of negacyclic BCH codes, the result
follows. The proof is complete.
We are now ready to show one of the main results of this subsection.
Theorem 7.5.2 Let n = q
2
+ 1, where q ≡ 1 mod 4 is a power of an odd prime
and suppose that s = n/2. Then there exists an (n, n − 2i + 1, 2; 1, 2i + 2) q 2 MDS
convolutional code, where 2 ≤ i ≤ n/2 − 1.
Proof First, note that gcd(n, q) = 1 and or d 2n (q
2
) = 2. Let β be a primitive 2nth
root of unity in F q 2m . Consider that C 2 is the negacyclic BCH code of length n over
F q 2 generated by the product of the minimal polynomials
C 2 = =g 2 (x) = =M
(s)
(x)M
(s+2)
(x) · · · · · M
(s+2i)
(x),
where 2 ≤ i ≤ s − 1.
By Theorem 7.5.1, a parity check matrix of C 2 is obtained from the matrix
H 2 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 β
s
β
2s
· · · β
(n−1)s
1 β
(s+2)
β
2(s+2)
· · · β
(n−1)(s+2)
1 β
(s+4)
β
2(s+4)
· · · β
(n−1)(s+4)
. . .
. . .
. . .
. . .
. . .
1 β
(s+2i)
β
2(s+2i)
· · · β
(n−1)(s+2i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) with respect to some
F q 2 -basis β of F q 4 and then removing one linearly dependent row. From Lemma 7.5.1,
this new matrix H C 2 has rank 2i + 1; so, C 2 has dimension n − 2i − 1. From the
BCH bound for negacyclic codes, it follows that the minimum distance d 2 of C 2
satisfies d 2 ≥ 2i + 2. Thus, from the (classical) Singleton bound, one concludes that
C 2 is a MDS code with parameters [n, n − 2i − 1, 2i + 2] q 2 and, consequently, its
Hermitian dual code has dimension 2i + 1.
Next we assume that C 1 is the negacyclic BCH code of length n over F q 2 generated
by the product of the minimal polynomials
C 1 = =g 1 (x) = =M
(s)
(x)M
(s+2)
(x) · · · · · M
[s+2(i−1)]
(x),
Similarly, by Theorem 7.5.1, C 1 has a parity check matrix derived from the matrix
H 1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1
β
s
β
2s
· · ·
β
(n−1)s
1 β
(s+2)
β
2(s+2)
· · · β
(n−1)(s+2)
1 β
(s+4)
β
2(s+4)
· · · β
(n−1)(s+4)
. . .
. . .
. . .
. . .
. . .
1 β
[s+2(i−1)]
β
2[s+2(i−1)]
· · · β
(n−1)[s+2(i−1)]
⎤
⎥
⎥
⎥
⎥
⎥
⎦