5.2 BCH Codes—Part II
87
Once the upper bound for the number of distinct q-ary cosets has been improved
(see Theorem 5.2.2), we are able to construct families q-ary CSS codes with good
parameters. Theorem 5.2.8 asserts the existence of such codes.
Theorem 5.2.8 Let n = q
m
− 1, where q ≥ 4 is a prime power and m ≥ 2 is an
even integer. Then there exists an [[n, n − 2m(c − 2) − m/2 − 1, d ≥ c]] q quantum
code, where 2 ≤ c ≤ q.
Proof Let C 1 be the cyclic code generated by
M
(0)
(x)M
(1)
(x) . . . M
(c−2)
(x),
and C 2 be the cyclic code generated by g 2 (x), that is the product of the minimal
polynomials
g 2 (x) =
i
M
(i)
(x),
where M
(i)
(x) are the minimal polynomials of α
i such that
i /
∈ {q
m/2
+ 1, q
m/2
+ 2, . . . , q
m/2
+ c − 1}.
From the BCH bound, the minimum distance of C 1 is greater than or equal to c,
since its defining set contains the sequence 0, 1, . . . , c − 2. Again, from the BCH
bound, the minimum distance of C
⊥
2 is also greater than or equal to c, because C
⊥
2 is
equivalent to C = =(x
n
− 1)/g 2 (x) and C contains the sequence q
m/2
+ 1, q
m/2
+
2, . . . , q
m/2
+ c − 1. From the CSS construction, the resulting quantum code has
minimum distance greater than or equal to c. Moreover, we have C 2 ⊂ C 1 .
The degree of g 1 (x) is equal to the cardinality of the defining set Z 1 of C 1 .
Moreover, from Theorem 5.2.7, Z 1 has m(c − 2) + 1 elements, so the dimension k 1
of C 1 is
k 1 = n − m(c − 2) − 1.
Similarly, by applying Theorem 5.2.7, since q
m/2
+ c − 1 < T := 2q
m/2 , then C 2
has dimension
k 2 = n − [n − (m(c − 2) + m/2)] = m(c − 2) + m/2.
Thus, the CSS code has dimension n − 2m(c − 2) − m/2 − 1. Therefore, an
[[n, n − 2m(c − 2) − m/2 − 1, d ≥ c]] q ,
quantum code can be constructed. The proof is complete.