7.3 Constructions of Optimal QCCs
169
7.3.1 Optimal Convolutional Codes
In this section, we construct a family of (classical) convolutional MDS codes that
will be utilized in the construction of quantum MDS convolutional codes exhibited
in the next subsection (Sect. 7.3.2).
We begin by recalling the following well-known result.
Lemma 7.3.2 ([114, Theorem 9, Chap. 11]) Suppose that q = 2
t , where t ≥ 2 is an
integer, n = q + 1 and consider that a =
q
2
. Then one has
(i) With exception of coset C 0 = {0}, each one of the other q-cosets is of the form
C a−i = {a − i, a + i + 1}, where 0 ≤ i ≤ a − 1;
(ii) The q-ary cosets C a−i = {a − i, a + i + 1}, where 0 ≤ i ≤ a − 1, are mutually
disjoint.
We are now able to construct optimal convolutional codes.
Theorem 7.3.2 Assume that q = 2
t , where t ≥ 3 is an integer, n = q + 1 and consider that a =
q
2
. Then there exists a classical MDS convolutional code with parameters (n, n − 2i, 2; 1, 2i + 3) q , where 1 ≤ i ≤ a − 1.
Proof We first note that gcd(n, q) = 1 and or d n (q) = 2. The proof consists of two
steps. We first construct suitable BCH (block) codes and after constructing convolutional codes derived from them.
Let C 2 be the BCH code of length n over F q generated by the product of the
minimal polynomials
C 2 = =g 2 (x) = =M
(a−i)
(x)M
(a−i+1)
(x) · · · · · M
(a−1)
(x)M
(a)
(x).
A parity check matrix of C 2 is obtained from the matrix
H 2i+3,a−i =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
. . .
. . .
. . .
. . .
. . .
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
1 α
a
· · ·
· · · α
(n−1)a
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) with respect to some
F q -basis β of F q 2 and then removing any linearly dependent rows. This new matrix
H C 2 is a parity check matrix of C 2 and it has 2i + 2 rows. Since the dimension of C 2
is equal to n − 2(i + 1) (as proved in the paragraph below), so there is no linearly
dependent rows in H C 2 .
From Lemma 7.3.2, each one of the q-ary cyclotomic cosets C a−i , where 0 ≤
i ≤ a − 1 (corresponding to the minimal polynomials M
(a−i)
(x)), has two elements
and they are mutually disjoint. Since the degree of the generator polynomial g 2 (x)
of the code C 2 equals the cardinality of its defining set, then one has deg(g 2 (x)) =
169
7.3.1 Optimal Convolutional Codes
In this section, we construct a family of (classical) convolutional MDS codes that
will be utilized in the construction of quantum MDS convolutional codes exhibited
in the next subsection (Sect. 7.3.2).
We begin by recalling the following well-known result.
Lemma 7.3.2 ([114, Theorem 9, Chap. 11]) Suppose that q = 2
t , where t ≥ 2 is an
integer, n = q + 1 and consider that a =
q
2
. Then one has
(i) With exception of coset C 0 = {0}, each one of the other q-cosets is of the form
C a−i = {a − i, a + i + 1}, where 0 ≤ i ≤ a − 1;
(ii) The q-ary cosets C a−i = {a − i, a + i + 1}, where 0 ≤ i ≤ a − 1, are mutually
disjoint.
We are now able to construct optimal convolutional codes.
Theorem 7.3.2 Assume that q = 2
t , where t ≥ 3 is an integer, n = q + 1 and consider that a =
q
2
. Then there exists a classical MDS convolutional code with parameters (n, n − 2i, 2; 1, 2i + 3) q , where 1 ≤ i ≤ a − 1.
Proof We first note that gcd(n, q) = 1 and or d n (q) = 2. The proof consists of two
steps. We first construct suitable BCH (block) codes and after constructing convolutional codes derived from them.
Let C 2 be the BCH code of length n over F q generated by the product of the
minimal polynomials
C 2 = =g 2 (x) = =M
(a−i)
(x)M
(a−i+1)
(x) · · · · · M
(a−1)
(x)M
(a)
(x).
A parity check matrix of C 2 is obtained from the matrix
H 2i+3,a−i =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
. . .
. . .
. . .
. . .
. . .
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
1 α
a
· · ·
· · · α
(n−1)a
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) with respect to some
F q -basis β of F q 2 and then removing any linearly dependent rows. This new matrix
H C 2 is a parity check matrix of C 2 and it has 2i + 2 rows. Since the dimension of C 2
is equal to n − 2(i + 1) (as proved in the paragraph below), so there is no linearly
dependent rows in H C 2 .
From Lemma 7.3.2, each one of the q-ary cyclotomic cosets C a−i , where 0 ≤
i ≤ a − 1 (corresponding to the minimal polynomials M
(a−i)
(x)), has two elements
and they are mutually disjoint. Since the degree of the generator polynomial g 2 (x)
of the code C 2 equals the cardinality of its defining set, then one has deg(g 2 (x)) =