5.3 BCH Codes—Part III
95
Table 5.11 Code comparison
Our CSS codes—Construction III
CSS Codes in [4]
[[n, n − m(2c − 3) − 1, d ≥ c]] q , 2 ≤ c ≤ q
[[n
, k
, d
]] q
m = 3
[[26, 16, d ≥ 3]] 3
[[26, 14, d
≥ 3]] 3
[[63, 53, d ≥ 3]] 4
[[63, 51, d
≥ 3]] 4
[[63, 47, d ≥ 4]] 4
[[63, 45, d
≥ 4]] 4
[[124, 114, d ≥ 3]] 5
[[124, 112, d
≥ 3]] 5
[[124, 108, d ≥ 4]] 5
[[124, 106, d
≥ 4]] 5
[[124, 102, d ≥ 5]] 5
[[124, 100, d
≥ 5]] 5
[[342, 332, d ≥ 3]] 7
[[342, 330, d
≥ 3]] 7
[[342, 326, d ≥ 4]] 7
[[342, 324, d
≥ 4]] 7
[[342, 320, d ≥ 5]] 7
[[342, 318, d
≥ 5]] 7
[[342, 314, d ≥ 6]] 7
[[342, 312, d
≥ 6]] 7
[[342, 308, d ≥ 7]] 7
[[342, 306, d
≥ 7]] 7
m = 4
[[255, 242, d ≥ 3]] 4
[[255, 239, d
≥ 3]] 4
[[255, 234, d ≥ 4]] 4
[[255, 231, d
≥ 4]] 4
[[624, 611, d ≥ 3]] 5
[[624, 608, d
≥ 3]] 5
[[624, 603, d ≥ 4]] 5
[[624, 600, d
≥ 4]] 5
[[624, 595, d ≥ 5]] 5
[[624, 592, d
≥ 5]] 5
5.3.1 Construction I
Let us prove the first result.
Lemma 5.3.4 Let n = q
4
− 1, where q ≥ 3 is a prime power, and consider the first
q
2
− 1 q
2 -ary cosets modulo n given by
C [q 2 +1] ,
C [q 2 +2] = {q
2
+ 2, 1 + 2q
2
},
. . .
C [2q 2 −1] = {2q
2
− 1, 1 + (q
2
− 1)q
2
}.
Then the following hold:
(a) C [q 2 +1] contains only one element;
(b) each of the other cosets contains two elements;
(c) each of these cosets are mutually disjoint.