5.1 BCH Codes—Part I
73
Table 5.1 displays a comparison of the parameters of some CSS codes constructed
here with the parameters of the CSS codes shown in [4], and Table 5.2 shows a comparison between our CSS codes with the quantum codes derived from the nonbinary
Steane’s construction (see Corollary 5.1.2).
Tables 5.3 and 5.4 show our codes obtained from Construction I and from Theorem 5.1.4 in Construction II, respectively. Table 5.5 presents some codes generated
from Construction III, and Table 5.6 displays some codes generated from Construction IV. Finally, Table 5.7 exhibited some codes derived from Construction V.
Checking the parameters of our quantum BCH codes tabulated, one can see that
our codes have parameters better than the ones available in the literature. In other
words, fixing the code length n and the minimum distance d (or the lower bound for
the minimum distance d, since the true minimum distance of BCH are not known
in general), the quantum BCH codes constructed here achieve greater values of the
number of qudits than the quantum BCH codes available in the literature.
Remark 5.1.2 The procedure of code comparison exhibited above will be adopted
throughout the entire book in order to perform the comparison among the parameters
of the quantum codes constructed here with the parameters of the quantum codes
available in the literature. In other words: to compare the parameters of an [n, k 1 , d] q
quantum code Q 1 constructed here, we perform searching for a quantum code of
length n and minimum distance d. If such code Q 2 is an [n, k 2 , d] q code with k 1 > k 2 ,
then Q 1 is better than Q 2 ; if k 2 > k 1 it implies that Q 2 is better than Q 1 . In many
cases we fix the code length and the lower bound for the minimum distance as it was
said above (see Tables 5.1, 5.2, 5.3, 5.4, 5.5, 5.6 and 5.7 to see this), after comparing
the code dimension, as was done earlier. This criterion of code comparison is usual
in the literature.
Note that our [[1093, 1079, d ≥ 3]] 3 code has the same parameters of the corresponding Hamming code; our [[71, 61, d ≥ 3]] 5 code can be compared with distance
three codes obtained by shortening Hamming codes.
The codes [[67, 61, d ≥ 3]] 29 and [[73, 67, d ≥ 3]] 64 shown in Table 5.7 have
parameters satisfying n + 2 − k − 2d ≤ 2; the parameters of the codes
[[11, 1, d ≥ 4]] 5 , [[35, 27, d ≥ 3]] 13 and [[35, 27, d ≥ 3]] 27 , satisfy n + 2 − k −
2d ≤ 4. The [[11, 1, d ≥ 4]] 5 code is comparable to the [[17, 9, 4]] 5 code shown in
[17], and the [[61, 51, d ≥ 3]] 9 code is comparable to the [[65, 51, 4]] 9 code shown
in [17].
Summarizing the results: in this subsection, we have presented five quantum code
constructions generating families of quantum BCH codes with good parameters.
73
Table 5.1 displays a comparison of the parameters of some CSS codes constructed
here with the parameters of the CSS codes shown in [4], and Table 5.2 shows a comparison between our CSS codes with the quantum codes derived from the nonbinary
Steane’s construction (see Corollary 5.1.2).
Tables 5.3 and 5.4 show our codes obtained from Construction I and from Theorem 5.1.4 in Construction II, respectively. Table 5.5 presents some codes generated
from Construction III, and Table 5.6 displays some codes generated from Construction IV. Finally, Table 5.7 exhibited some codes derived from Construction V.
Checking the parameters of our quantum BCH codes tabulated, one can see that
our codes have parameters better than the ones available in the literature. In other
words, fixing the code length n and the minimum distance d (or the lower bound for
the minimum distance d, since the true minimum distance of BCH are not known
in general), the quantum BCH codes constructed here achieve greater values of the
number of qudits than the quantum BCH codes available in the literature.
Remark 5.1.2 The procedure of code comparison exhibited above will be adopted
throughout the entire book in order to perform the comparison among the parameters
of the quantum codes constructed here with the parameters of the quantum codes
available in the literature. In other words: to compare the parameters of an [n, k 1 , d] q
quantum code Q 1 constructed here, we perform searching for a quantum code of
length n and minimum distance d. If such code Q 2 is an [n, k 2 , d] q code with k 1 > k 2 ,
then Q 1 is better than Q 2 ; if k 2 > k 1 it implies that Q 2 is better than Q 1 . In many
cases we fix the code length and the lower bound for the minimum distance as it was
said above (see Tables 5.1, 5.2, 5.3, 5.4, 5.5, 5.6 and 5.7 to see this), after comparing
the code dimension, as was done earlier. This criterion of code comparison is usual
in the literature.
Note that our [[1093, 1079, d ≥ 3]] 3 code has the same parameters of the corresponding Hamming code; our [[71, 61, d ≥ 3]] 5 code can be compared with distance
three codes obtained by shortening Hamming codes.
The codes [[67, 61, d ≥ 3]] 29 and [[73, 67, d ≥ 3]] 64 shown in Table 5.7 have
parameters satisfying n + 2 − k − 2d ≤ 2; the parameters of the codes
[[11, 1, d ≥ 4]] 5 , [[35, 27, d ≥ 3]] 13 and [[35, 27, d ≥ 3]] 27 , satisfy n + 2 − k −
2d ≤ 4. The [[11, 1, d ≥ 4]] 5 code is comparable to the [[17, 9, 4]] 5 code shown in
[17], and the [[61, 51, d ≥ 3]] 9 code is comparable to the [[65, 51, 4]] 9 code shown
in [17].
Summarizing the results: in this subsection, we have presented five quantum code
constructions generating families of quantum BCH codes with good parameters.