86
5 Quantum Code Constructions
Proof It suffices to consider C 1 and C 2 as the cyclic codes generated, respectively,
by
C 1 = =M
(0)
(x)M
(1)
(x)M
(2)
(x) . . . M
(q−2)
(x)
and
C 2 =
i
M
(i)
(x)
,
where M
(i)
(x) are all minimal polynomials of α
i such that
i /
∈ {q + 1, q + 2, . . . , 2q − 1}.
Proceeding similarly as in the proof of Theorem 5.2.5 changing p for q, the result
follows.
Corollary 5.2.5 There exists an [[q
2
− 1, q
2
− 4c + 5, d ≥ c]] q quantum code,
where c < q, q = p
m , and q ≥ 4.
Proof Let C 1 and C 2 be cyclic code generated, respectively, by
C 1 = =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 /
∈ {q + 1, q + 2, . . . , q + (c − 1)}.
Proceeding similarly as in the proof of Corollary 5.2.4 changing p for q, the result
follows.
5.2.2.2 Construction II
We need to utilize the following result shown in [166] in order to obtain our families
of quantum codes.
Theorem 5.2.7 [166] | C s |= m for all 0 < s < T := 2q
m/2 except | C q m/2 +1 |=
m/2 when m is even.