5.2 BCH Codes—Part II
77
Properties of cyclotomic cosets have been studied in many areas of research,
especially in theory of classical and quantum error-correcting codes. For instance,
the q-cosets were investigated by several researchers in order to obtain efficient
classical cyclic codes [15, 110, 114, 115, 129, 141–143, 165, 166], as well as to
construct efficient quantum codes [4, 17, 25, 30, 56, 80, 81, 89, 108, 113, 148, 153,
155, 156, 162, 163].
In this subsection we show properties of q-ary cosets modulo n = q
m
− 1, where
q = 2 in order to construct families of good quantum BCH codes by applying the
CSS construction. Our codes have parameters
• [[q
2
− 1, q
2
− 4c + 5, d ≥ c]] q ,
where 2 ≤ c ≤ q and q ≥ 4 is a prime power;
• [[n, n − 2m(c − 2) − m/2 − 1, d ≥ c]] q ,
where n = q
m
− 1, q ≥ 4, 2 ≤ c ≤ q and m ≥ 2 is an even integer;
• [[q
m
− 1, q
m
− 6m − 3, d ≥ 5]] q ,
• [[q
m
− 1, q
m
− 2m − 3, d ≥ 3]] q ,
• [[q
m
− 1, q
m
− 4m − 3, d ≥ 4]] q ,
where q ≥ 5 is an odd prime power;
• [[n, n − m(2c − 3) − 1, d ≥ c]] q ,
where n = q
m
− 1, q ≥ 3 is a prime power, m ≥ 3 and 2 ≤ c ≤ q.
To the reader’s convenience, we present a brief organization of the topics.
Section 5.2.1 establishes properties concerning q-cosets modulo q
m
− 1. These
results will be applied for constructing q-ary CSS quantum codes with good parameters. In Sect. 5.2.2, we explain how to obtain families of CSS codes by utilizing such
properties. In Sect. 5.2.3, the parameters of our CSS codes are compared with the
ones available in the literature.
5.2.1 Properties of q-cosets
We here explore the structure of q-ary cosets in order to show nice properties of them.
As already said, we are interested in the study of q-ary cosets modulo n = q
m
− 1.
Proposition 5.2.1 Let q be an odd prime power and C s be a q-ary coset with representative s. Then s is even if and only if ∀ t ∈ C s , t is even.
Proof Suppose first that s = 2k, where k is an integer and let t be an element of the
coset C s = {s, qs, q
2 s, q
3 s, . . . , q
m s −1 s} without considering the modulo operation.
Then it follows that t = sq
l
= 2kq
l , where 0 ≤ l ≤ m s − 1. Applying the division
algorithm for t and q
m
− 1, one has 2kq
l
= (q
m
− 1)a + r , where r is an integer such
that 0 ≤ r < q
m
− 1. Hence, it follows that r = 2kq
l
− (q
m
− 1)a. Since q
m
− 1 is
even, r is also even, as required.
Conversely, suppose that each t, where t ∈ C s , (considering the modulo operation)
is of the form t = 2k, with k integer. Applying again the division algorithm for sq
l
77
Properties of cyclotomic cosets have been studied in many areas of research,
especially in theory of classical and quantum error-correcting codes. For instance,
the q-cosets were investigated by several researchers in order to obtain efficient
classical cyclic codes [15, 110, 114, 115, 129, 141–143, 165, 166], as well as to
construct efficient quantum codes [4, 17, 25, 30, 56, 80, 81, 89, 108, 113, 148, 153,
155, 156, 162, 163].
In this subsection we show properties of q-ary cosets modulo n = q
m
− 1, where
q = 2 in order to construct families of good quantum BCH codes by applying the
CSS construction. Our codes have parameters
• [[q
2
− 1, q
2
− 4c + 5, d ≥ c]] q ,
where 2 ≤ c ≤ q and q ≥ 4 is a prime power;
• [[n, n − 2m(c − 2) − m/2 − 1, d ≥ c]] q ,
where n = q
m
− 1, q ≥ 4, 2 ≤ c ≤ q and m ≥ 2 is an even integer;
• [[q
m
− 1, q
m
− 6m − 3, d ≥ 5]] q ,
• [[q
m
− 1, q
m
− 2m − 3, d ≥ 3]] q ,
• [[q
m
− 1, q
m
− 4m − 3, d ≥ 4]] q ,
where q ≥ 5 is an odd prime power;
• [[n, n − m(2c − 3) − 1, d ≥ c]] q ,
where n = q
m
− 1, q ≥ 3 is a prime power, m ≥ 3 and 2 ≤ c ≤ q.
To the reader’s convenience, we present a brief organization of the topics.
Section 5.2.1 establishes properties concerning q-cosets modulo q
m
− 1. These
results will be applied for constructing q-ary CSS quantum codes with good parameters. In Sect. 5.2.2, we explain how to obtain families of CSS codes by utilizing such
properties. In Sect. 5.2.3, the parameters of our CSS codes are compared with the
ones available in the literature.
5.2.1 Properties of q-cosets
We here explore the structure of q-ary cosets in order to show nice properties of them.
As already said, we are interested in the study of q-ary cosets modulo n = q
m
− 1.
Proposition 5.2.1 Let q be an odd prime power and C s be a q-ary coset with representative s. Then s is even if and only if ∀ t ∈ C s , t is even.
Proof Suppose first that s = 2k, where k is an integer and let t be an element of the
coset C s = {s, qs, q
2 s, q
3 s, . . . , q
m s −1 s} without considering the modulo operation.
Then it follows that t = sq
l
= 2kq
l , where 0 ≤ l ≤ m s − 1. Applying the division
algorithm for t and q
m
− 1, one has 2kq
l
= (q
m
− 1)a + r , where r is an integer such
that 0 ≤ r < q
m
− 1. Hence, it follows that r = 2kq
l
− (q
m
− 1)a. Since q
m
− 1 is
even, r is also even, as required.
Conversely, suppose that each t, where t ∈ C s , (considering the modulo operation)
is of the form t = 2k, with k integer. Applying again the division algorithm for sq
l