64
5 Quantum Code Constructions
are mutually disjoint, it follows that C 2 C 1 . Applying the CSS construction to
C 1 and C 2 , one obtains an [[n, n − 2mr, d ≥ r + 2]] q quantum code, and we are
done.
Example 5.1.2 Theorem 5.1.4 has variants as follows: to construct an [19, 13, d ≥
3]] 7 quantum code, let us consider that q = 7, n = 19 and m = 3. The cosets are
given by C 2 = {2, 14, 3} and C 16 = {5, 16, 17}. Let C 1 be generated by M
(2)
(x) and
C 2 generated by g 2 (x) =
i
M
(i)
(x), where i /
∈ {16} and i runs through the coset
representatives mod 19. Then an [[19, 13, d ≥ 3]] 7 quantum code can be constructed.
Proceeding similarly, one can get quantum codes with parameters [[31, 25, d ≥ 3]] 5 ,
[[71, 61, d ≥ 3]] 5 , [[11, 1, d ≥ 4]] 3 , [[31, 19, d ≥ 4]] 5 , [[31, 13, d ≥ 5]] 5 ,
[[71, 51, d ≥ 4]] 5 , [[71, 41, d ≥ 6]] 5 .
5.1.3 Construction III
In this subsection, we construct families of quantum BCH codes of prime length
by applying Steane’s enlargement of nonbinary CSS construction [62, Corollary
4]. These new families have parameters better than the parameters of the quantum
BCH codes available in the literature. Let us recall the Steane enlargement code
construction applied to nonbinary alphabets.
Corollary 5.1.2 ([62, Corollary 4]) Assume that we have an [N 0 , K 0 ] linear code
L which contains its Euclidean dual, L
⊥
≤ L, and which can be enlarged to an
[N 0 , K
0 ] linear code L
, where K
0 ≥ K 0 + 2. Then there exists a quantum code with
parameters [[N 0 , K 0 + K
0 − N 0 , d ≥ min{d,
q+1
q
d
}]], where d = w(L\L
⊥ )
and d
= w(L
\L
⊥ ).
Euclidean dual-containing cyclic codes can be derived from Lemma 5.1.3.
Lemma 5.1.3 [4, Lemma 1] Assume that gcd(q, n) = 1. A cyclic code of length n
over F q with defining set Z contains its Euclidean dual code if and only if Z ∩ Z
−1
=
∅, where Z
−1
= {−z mod n | z ∈ Z }.
In Lemma 5.1.2 of Sect. 5.1.2 we have shown the existence of, at least, one qary cyclotomic coset containing two consecutive integers provided the code length
is prime. In what follows, we show how to construct good quantum codes of prime
length by applying Steane’s code construction. We begin by presenting an illustrative
example.
Example 5.1.3 Assume that n = 31 and q = 5. From Lemma 5.1.2, there exists a
coset containing at least two consecutive integers; here it is the coset C 8 = {8, 9, 14}.
Let C be the cyclic code generated by the product of the minimal polynomials C =
g(x) = =M
(4)
(x)M
(8)
(x). C has defining set Z = C 4 ∪ C 8 = {4, 7, 8, 9, 14, 20}
5 Quantum Code Constructions
are mutually disjoint, it follows that C 2 C 1 . Applying the CSS construction to
C 1 and C 2 , one obtains an [[n, n − 2mr, d ≥ r + 2]] q quantum code, and we are
done.
Example 5.1.2 Theorem 5.1.4 has variants as follows: to construct an [19, 13, d ≥
3]] 7 quantum code, let us consider that q = 7, n = 19 and m = 3. The cosets are
given by C 2 = {2, 14, 3} and C 16 = {5, 16, 17}. Let C 1 be generated by M
(2)
(x) and
C 2 generated by g 2 (x) =
i
M
(i)
(x), where i /
∈ {16} and i runs through the coset
representatives mod 19. Then an [[19, 13, d ≥ 3]] 7 quantum code can be constructed.
Proceeding similarly, one can get quantum codes with parameters [[31, 25, d ≥ 3]] 5 ,
[[71, 61, d ≥ 3]] 5 , [[11, 1, d ≥ 4]] 3 , [[31, 19, d ≥ 4]] 5 , [[31, 13, d ≥ 5]] 5 ,
[[71, 51, d ≥ 4]] 5 , [[71, 41, d ≥ 6]] 5 .
5.1.3 Construction III
In this subsection, we construct families of quantum BCH codes of prime length
by applying Steane’s enlargement of nonbinary CSS construction [62, Corollary
4]. These new families have parameters better than the parameters of the quantum
BCH codes available in the literature. Let us recall the Steane enlargement code
construction applied to nonbinary alphabets.
Corollary 5.1.2 ([62, Corollary 4]) Assume that we have an [N 0 , K 0 ] linear code
L which contains its Euclidean dual, L
⊥
≤ L, and which can be enlarged to an
[N 0 , K
0 ] linear code L
, where K
0 ≥ K 0 + 2. Then there exists a quantum code with
parameters [[N 0 , K 0 + K
0 − N 0 , d ≥ min{d,
q+1
q
d
}]], where d = w(L\L
⊥ )
and d
= w(L
\L
⊥ ).
Euclidean dual-containing cyclic codes can be derived from Lemma 5.1.3.
Lemma 5.1.3 [4, Lemma 1] Assume that gcd(q, n) = 1. A cyclic code of length n
over F q with defining set Z contains its Euclidean dual code if and only if Z ∩ Z
−1
=
∅, where Z
−1
= {−z mod n | z ∈ Z }.
In Lemma 5.1.2 of Sect. 5.1.2 we have shown the existence of, at least, one qary cyclotomic coset containing two consecutive integers provided the code length
is prime. In what follows, we show how to construct good quantum codes of prime
length by applying Steane’s code construction. We begin by presenting an illustrative
example.
Example 5.1.3 Assume that n = 31 and q = 5. From Lemma 5.1.2, there exists a
coset containing at least two consecutive integers; here it is the coset C 8 = {8, 9, 14}.
Let C be the cyclic code generated by the product of the minimal polynomials C =
g(x) = =M
(4)
(x)M
(8)
(x). C has defining set Z = C 4 ∪ C 8 = {4, 7, 8, 9, 14, 20}