5.2 BCH Codes—Part II
89
Since 6q
i
≡ 2 mod n does not hold, the result follows.
It is easy to see that each one of the cosets C (
q 3 −1
2 +1)
, C (
q 3 −1
2 +2)
and C (
q 3 −1
2 +3)
has
three elements.
Let C 1 be the cyclic code generated by
M
(0)
(x)M
(1)
(x)M
(2)
(x)M
(3)
(x)
and C 2 be the cyclic code generated by
i
M
(i)
(x),
where M
(i)
(x) are the minimal polynomials such that i /
∈ {b, b + 1, b + 2, b + 3}
and i runs through the coset representatives mod (q
3
− 1) and b =
q
3 −1
2
. Proceeding
similarly as in the proof of Theorem 5.2.5 and applying the CSS construction, the
code follows.
Applying the previous theorem one can construct quantum codes with parameters
[[124, 104, d ≥ 5]] 5 , [[342, 322, d ≥ 5]] 7 , [[1330, 1310, d ≥ 5]] 11 , and so on.
We can also construct CSS codes with minimum distance greater than three and
four, as states the next result.
Corollary 5.2.6 Let q ≥ 5 be an odd prime power. Then there exist CSS codes with
parameters [[q
3
− 1, q
3
− 9, d ≥ 3]] q and [[q
3
− 1, q
3
− 15, d ≥ 4]] q .
Proof For the first construction, consider C 1 be the cyclic code generated by
M
(0)
(x)M
(1)
(x)
and C 2 be the cyclic code generated by
i
M
(i)
(x),
where M
(i) are the minimal polynomials such that i /
∈ {b, b + 1} and i runs through
the coset representatives mod (q
3
− 1) and b =
q
3 −1
2
.
For the second, let us consider C 1 as the cyclic code generated by
M
(0)
(x)M
(1)
(x)M
(2)
(x)
and C 2 be the cyclic code generated by
i
M
(i)
(x),