5.2 BCH Codes—Part II
91
where 2 ≤ c ≤ q, and C 2 be the cyclic code generated by
i
M
(i)
(x),
where M
(i)
(x) are the minimal polynomials such that i /
∈ {q + 1, 2q + 1, . . . , (c −
1)q + 1}. Applying Lemma 5.2.3 and Theorem 5.2.4, and proceeding similarly as in
the proof of Theorem 5.2.5 the result follows.
Example 5.2.1 Applying Theorem 5.2.11 with q = 3, n = 3
3
− 1 = 26 and d ≥ 3
one can construct an [[26, 16, d ≥ 3]] 3 quantum code. If q = 4, n = 63 and d ≥ 3
one has an [[63, 53, d ≥ 3]] 4 quantum code. Similarly, if we take q = 5, n = 124
and d ≥ 4 one has an [[124, 108, d ≥ 4]] 5 quantum code and so on.
5.2.3 Code Comparison
In this section we compare the parameters of our CSS codes with the parameters
of the best CSS codes shown in [4]. As was said, we utilize the code comparison
described in Remark 5.1.2. Such criterion is usual in the literature.
In Table 5.8, the parameters [[n, k, d ≥ c]] q assume the values [[q
2
− 1, q
2
− 4c+
5, d ≥ c]] q , where 2 ≤ c ≤ q and q ≥ 4 is a prime power.
In Table 5.9, n = q
m
− 1 is the code length, where q ≥ 4 and m ≥ 2 is an even
integer; k = n − 2m(c − 2) − m/2 − 1, where 2 ≤ c ≤ q and d is the minimum distance of the respective code. The parameters of our codes are obtained from Construction I. The parameters [[n
, k
, d
]] q = [[n, n − 2m((δ − 1)(1 − 1/q)), d ≥ δ]] q .
are the parameters of the codes shown in [4].
In Table 5.10, our codes are derived from Construction II, where q ≥ 5 is an odd
prime power and [[n
, k
, d
]] q assumes the values mentioned above. In Table 5.11, the
codes are derived from Construction III, where n = q
m
− 1, q ≥ 3 is a prime power,
m ≥ 3 and 2 ≤ c ≤ q. The parameters [[n
, k
, d
]] q assume the values mentioned
above.
As can be seen in Tables 5.8, 5.9, 5.10 and 5.11, our CSS codes have parameters
better than the ones available in [4]. More precisely, fixing n and d, the codes constructed here achieve greater values of the number of qudits than the codes shown in
[4].
5.3 BCH Codes—Part III
In this section, we construct more families of quantum codes derived from BCH
codes. The codes constructed here can be found in our paper [89].
91
where 2 ≤ c ≤ q, and C 2 be the cyclic code generated by
i
M
(i)
(x),
where M
(i)
(x) are the minimal polynomials such that i /
∈ {q + 1, 2q + 1, . . . , (c −
1)q + 1}. Applying Lemma 5.2.3 and Theorem 5.2.4, and proceeding similarly as in
the proof of Theorem 5.2.5 the result follows.
Example 5.2.1 Applying Theorem 5.2.11 with q = 3, n = 3
3
− 1 = 26 and d ≥ 3
one can construct an [[26, 16, d ≥ 3]] 3 quantum code. If q = 4, n = 63 and d ≥ 3
one has an [[63, 53, d ≥ 3]] 4 quantum code. Similarly, if we take q = 5, n = 124
and d ≥ 4 one has an [[124, 108, d ≥ 4]] 5 quantum code and so on.
5.2.3 Code Comparison
In this section we compare the parameters of our CSS codes with the parameters
of the best CSS codes shown in [4]. As was said, we utilize the code comparison
described in Remark 5.1.2. Such criterion is usual in the literature.
In Table 5.8, the parameters [[n, k, d ≥ c]] q assume the values [[q
2
− 1, q
2
− 4c+
5, d ≥ c]] q , where 2 ≤ c ≤ q and q ≥ 4 is a prime power.
In Table 5.9, n = q
m
− 1 is the code length, where q ≥ 4 and m ≥ 2 is an even
integer; k = n − 2m(c − 2) − m/2 − 1, where 2 ≤ c ≤ q and d is the minimum distance of the respective code. The parameters of our codes are obtained from Construction I. The parameters [[n
, k
, d
]] q = [[n, n − 2m((δ − 1)(1 − 1/q)), d ≥ δ]] q .
are the parameters of the codes shown in [4].
In Table 5.10, our codes are derived from Construction II, where q ≥ 5 is an odd
prime power and [[n
, k
, d
]] q assumes the values mentioned above. In Table 5.11, the
codes are derived from Construction III, where n = q
m
− 1, q ≥ 3 is a prime power,
m ≥ 3 and 2 ≤ c ≤ q. The parameters [[n
, k
, d
]] q assume the values mentioned
above.
As can be seen in Tables 5.8, 5.9, 5.10 and 5.11, our CSS codes have parameters
better than the ones available in [4]. More precisely, fixing n and d, the codes constructed here achieve greater values of the number of qudits than the codes shown in
[4].
5.3 BCH Codes—Part III
In this section, we construct more families of quantum codes derived from BCH
codes. The codes constructed here can be found in our paper [89].