58
5 Quantum Code Constructions
non-narrow-sense BCH codes constructed here. The fifth and last construction is
based on finding cyclic codes whose defining set consists of only one coset with
at least two consecutive integers. The content presented in Sect. 5.1.1–5.1.4 can be
found in our paper [100] and the material of Sect. 5.1.5 is based on our paper [103].
The key of most of these constructions is the investigation of suitable properties of
cyclotomic cosets. More precisely, we need to know the cardinality of each of them
and also (when it is possible) to compute a large quantity of consecutive elements in
the union of all cosets (the defining set) of a given cyclic code C in order to obtain the
exact dimension of the code as well as a lower bound for the minimum distance of
it. In other words, knowing the cardinality of the defining set Z of a given BCH code
C and by computing the maximum sequence of consecutive integers belonging to Z
we know the dimension and the maximum lower bound for the minimum distance
of C, respectively.
Since until now the true minimum distance of BCH code is not known and since
the computation of its dimension in all cases is also not known, this is an interesting
area of research. In other words, there is much challenge to be surpassed.
As we will see in the sequence, in a particular case, we apply the concept of linear
congruence, to prove (for codes of prime length) the existence of at least one q-ary
coset containing two consecutive integers. This is interesting because by means of this
result we can construct families of nonbinary quantum codes with good parameters.
To be more precise, our families of quantum BCH codes have parameters described
in the following sequence:
• [[n, n − 4(c − 2) − 2, d ≥ c]] q , where q ≥ 4 is a prime power, n is an integer with
gcd(q, n) = 1, (q − 1) | n, m = ord n (q) = 2 and 2 ≤ c ≤ r , where r is such that
n = r (q − 1);
• [[n, n − 2mr, d ≥ r + 2]] q , where m = ord n (q) ≥ 2, n is a prime number and r
is the number of cosets satisfying suitable conditions (see Theorem 5.1.4);
• [[n, n − m(2r − 1), d ≥ r + 2]] q , where m = ord n (q) ≥ 2, n is a prime number
and q ≥ 3;
• [[n, n − 4c, d ≥ c + 2]] q , where n > q is an integer with gcd(q, n) = 1, (q − 1) |
n, m = ord n (q) = 2, 1 ≤ c ≤ r − 3 and r > 3 satisfies n = r (q − 1);
• [[n, n − 4c − 2, d ≥ c + 2]] q , where 2 ≤ c ≤ r − 2, q > 3, n = r (q
2
− 1), r > 1
and m = ord n (q
2
) = 2;
• [[n, n − 2mr, d ≥ r + 2]] q , where q ≥ 3 is a prime power, n > q
2 is a prime
number such that gcd(q, n) = 1, m = ord n (q
2
) ≥ 2 and r is the number of cosets
satisfying suitable conditions (see Theorem 5.1.9).
• [[n, n − 2m
∗
, d ≥ r + 2]] q , where q ≥ 3 is a prime power and n > m (m =
ord n (q) ≥ r + 2) is a positive integer such that gcd(q, n) = 1, gcd(q
a i − 1, n) =
1 for every i = 1, 2, . . . , r , where 1 ≤ r, a 1 , a 2 , . . . , a r < m are integers, and
n| gcd(t 2 , . . . , t r ), where t j = [( j − ( j − 1)q
a j )(q
a j − 1)
−1
− (q
a 1 − 1)
−1
] for
every j = 2, . . . , r (the operations are performed modulo n).
Before proceeding further, we communicate to the reader that we will utilize
freely the expressions such as q-ary coset, q-ary coset, q-coset, or even coset when
the context is clear, meaning, of course, a cyclotomic coset. Another important remark
5 Quantum Code Constructions
non-narrow-sense BCH codes constructed here. The fifth and last construction is
based on finding cyclic codes whose defining set consists of only one coset with
at least two consecutive integers. The content presented in Sect. 5.1.1–5.1.4 can be
found in our paper [100] and the material of Sect. 5.1.5 is based on our paper [103].
The key of most of these constructions is the investigation of suitable properties of
cyclotomic cosets. More precisely, we need to know the cardinality of each of them
and also (when it is possible) to compute a large quantity of consecutive elements in
the union of all cosets (the defining set) of a given cyclic code C in order to obtain the
exact dimension of the code as well as a lower bound for the minimum distance of
it. In other words, knowing the cardinality of the defining set Z of a given BCH code
C and by computing the maximum sequence of consecutive integers belonging to Z
we know the dimension and the maximum lower bound for the minimum distance
of C, respectively.
Since until now the true minimum distance of BCH code is not known and since
the computation of its dimension in all cases is also not known, this is an interesting
area of research. In other words, there is much challenge to be surpassed.
As we will see in the sequence, in a particular case, we apply the concept of linear
congruence, to prove (for codes of prime length) the existence of at least one q-ary
coset containing two consecutive integers. This is interesting because by means of this
result we can construct families of nonbinary quantum codes with good parameters.
To be more precise, our families of quantum BCH codes have parameters described
in the following sequence:
• [[n, n − 4(c − 2) − 2, d ≥ c]] q , where q ≥ 4 is a prime power, n is an integer with
gcd(q, n) = 1, (q − 1) | n, m = ord n (q) = 2 and 2 ≤ c ≤ r , where r is such that
n = r (q − 1);
• [[n, n − 2mr, d ≥ r + 2]] q , where m = ord n (q) ≥ 2, n is a prime number and r
is the number of cosets satisfying suitable conditions (see Theorem 5.1.4);
• [[n, n − m(2r − 1), d ≥ r + 2]] q , where m = ord n (q) ≥ 2, n is a prime number
and q ≥ 3;
• [[n, n − 4c, d ≥ c + 2]] q , where n > q is an integer with gcd(q, n) = 1, (q − 1) |
n, m = ord n (q) = 2, 1 ≤ c ≤ r − 3 and r > 3 satisfies n = r (q − 1);
• [[n, n − 4c − 2, d ≥ c + 2]] q , where 2 ≤ c ≤ r − 2, q > 3, n = r (q
2
− 1), r > 1
and m = ord n (q
2
) = 2;
• [[n, n − 2mr, d ≥ r + 2]] q , where q ≥ 3 is a prime power, n > q
2 is a prime
number such that gcd(q, n) = 1, m = ord n (q
2
) ≥ 2 and r is the number of cosets
satisfying suitable conditions (see Theorem 5.1.9).
• [[n, n − 2m
∗
, d ≥ r + 2]] q , where q ≥ 3 is a prime power and n > m (m =
ord n (q) ≥ r + 2) is a positive integer such that gcd(q, n) = 1, gcd(q
a i − 1, n) =
1 for every i = 1, 2, . . . , r , where 1 ≤ r, a 1 , a 2 , . . . , a r < m are integers, and
n| gcd(t 2 , . . . , t r ), where t j = [( j − ( j − 1)q
a j )(q
a j − 1)
−1
− (q
a 1 − 1)
−1
] for
every j = 2, . . . , r (the operations are performed modulo n).
Before proceeding further, we communicate to the reader that we will utilize
freely the expressions such as q-ary coset, q-ary coset, q-coset, or even coset when
the context is clear, meaning, of course, a cyclotomic coset. Another important remark