5.2 BCH Codes—Part II
85
than or equal to p. Since the code C
⊥
2 is equivalent to C, C
⊥
2 also has minimum distance greater than or equal to p. Therefore, the resulting CSS code has minimum
distance greater than or equal to p.
Next we compute the dimension of the new CSS codes. We know that the
defining set Z 1 of C 1 has p − 1 disjoint cosets. Moreover, from Lemma 5.2.4,
all of them (except coset) C 0 have two elements. We know that the degree of
the generator polynomial of a cyclic code is equal to the cardinality of its defining set. Hence, C 1 has dimension k 1 = n − ∂(g 1 (x)), where n = p
2
− 1. From
Lemma 5.2.4, Z 1 has 2( p − 2) + 1 = 2 p − 3 elements, so k 1 = p
2
− 2 p + 2. Similarly, the dimension k 2 of C 2 is equal to k 2 = 2 p − 3. Thus, the CSS code has
dimension k 1 − k 2 = p
2
− 4 p + 5. Applying the CSS construction to C 1 and C 2 ,
we have an [[ p
2
− 1, p
2
− 4 p + 5, d ≥ p]] p CSS code.
Proceeding analogously as in the proof of Theorem 5.2.5, we obtain more families
of quantum codes.
Corollary 5.2.4 There exists an [[ p
2
− 1, p
2
− 4c + 5, d ≥ c]] p quantum code,
where c < p, and p is prime.
Proof Let C 1 and C 2 be two cyclic codes generated, respectively, by
M
(0)
(x)M
(1)
(x)M
(2)
(x) . . . M
(c−2)
(x)
and
C 2 =
i
M
(i)
(x)
,
where M
(i)
(x) are all minimal polynomials of α
i such that
i /
∈ {p + 1, p + 2, . . . , p + (c − 1)}.
Proceeding similarly as in the proof of Theorem 5.2.5, we have an
[[ p
2
− 1, p
2
− 4c + 5, d ≥ c]] p
quantum code.
Remark 5.2.3 It is clear that the previous code constructions also hold when considering q = p
m instead of considering p ( p prime), and F q l (l ≥ 2) instead of
considering F q , since the properties of cosets and the minimal polynomials are the
same when considered over F p as well as over F q .
Based on Remark 5.2.3, Theorem 5.2.5 and Corollary 5.2.4 can be easily extended
for all prime power q.
Theorem 5.2.6 Let q ≥ 4 be a prime power and n = q
2
− 1. Then, there exist CSS
codes with parameters [[q
2
− 1, q
2
− 4q + 5, d ≥ q]] q .
85
than or equal to p. Since the code C
⊥
2 is equivalent to C, C
⊥
2 also has minimum distance greater than or equal to p. Therefore, the resulting CSS code has minimum
distance greater than or equal to p.
Next we compute the dimension of the new CSS codes. We know that the
defining set Z 1 of C 1 has p − 1 disjoint cosets. Moreover, from Lemma 5.2.4,
all of them (except coset) C 0 have two elements. We know that the degree of
the generator polynomial of a cyclic code is equal to the cardinality of its defining set. Hence, C 1 has dimension k 1 = n − ∂(g 1 (x)), where n = p
2
− 1. From
Lemma 5.2.4, Z 1 has 2( p − 2) + 1 = 2 p − 3 elements, so k 1 = p
2
− 2 p + 2. Similarly, the dimension k 2 of C 2 is equal to k 2 = 2 p − 3. Thus, the CSS code has
dimension k 1 − k 2 = p
2
− 4 p + 5. Applying the CSS construction to C 1 and C 2 ,
we have an [[ p
2
− 1, p
2
− 4 p + 5, d ≥ p]] p CSS code.
Proceeding analogously as in the proof of Theorem 5.2.5, we obtain more families
of quantum codes.
Corollary 5.2.4 There exists an [[ p
2
− 1, p
2
− 4c + 5, d ≥ c]] p quantum code,
where c < p, and p is prime.
Proof Let C 1 and C 2 be two cyclic codes generated, respectively, by
M
(0)
(x)M
(1)
(x)M
(2)
(x) . . . M
(c−2)
(x)
and
C 2 =
i
M
(i)
(x)
,
where M
(i)
(x) are all minimal polynomials of α
i such that
i /
∈ {p + 1, p + 2, . . . , p + (c − 1)}.
Proceeding similarly as in the proof of Theorem 5.2.5, we have an
[[ p
2
− 1, p
2
− 4c + 5, d ≥ c]] p
quantum code.
Remark 5.2.3 It is clear that the previous code constructions also hold when considering q = p
m instead of considering p ( p prime), and F q l (l ≥ 2) instead of
considering F q , since the properties of cosets and the minimal polynomials are the
same when considered over F p as well as over F q .
Based on Remark 5.2.3, Theorem 5.2.5 and Corollary 5.2.4 can be easily extended
for all prime power q.
Theorem 5.2.6 Let q ≥ 4 be a prime power and n = q
2
− 1. Then, there exist CSS
codes with parameters [[q
2
− 1, q
2
− 4q + 5, d ≥ q]] q .