74
5 Quantum Code Constructions
Table 5.1 Code comparison
Our CSS codes
CSS codes in [4]
[[n, k, d]] q
[[n
, k
, d
]] q
[[40, 30, d ≥ 4]] 9
[[40, 28, d
≥ 4]] 9
[[40, 20, d ≥ 7]] 9
—
[[30, 7, d ≥ 8]] 11
—
[[61, 55, d ≥ 3]] 13
[[61, 49, d
≥ 3]] 13
[[84, 74, d ≥ 4]] 13
[[84, 72, d
≥ 4]] 13
[[84, 70, d ≥ 5]] 13
[[84, 68, d
≥ 5]] 13
[[84, 66, d ≥ 6]] 13
[[84, 64, d
≥ 6]] 13
[[91, 85, d ≥ 3]] 16
[[91, 79, d
≥ 3]] 16
[[144, 126, d ≥ 6]] 17
[[144, 124, d
≥ 6]] 17
[[144, 122, d ≥ 7]] 17
[[144, 120, d
≥ 7]] 17
[[144, 118, d ≥ 8]] 17
[[144, 116, d
≥ 8]] 17
[[127, 121, d ≥ 3]] 19
[[127, 115, d
≥ 3]] 19
Table 5.2 Code comparison
Our CSS codes
q-ary Steane’s construction
[[n, k, d]] q
[[n
, k
, d
]] q
[[19, 13, d ≥ 3]] 7
—
[[13, 7, d ≥ 3]] 9
—
[[19, 13, d ≥ 3]] 11
—
[[61, 55, d ≥ 3]] 13
[[61, 52, d
≥ 3]] 13
[[91, 85, d ≥ 3]] 16
[[91, 82, d
≥ 3]] 16
[[127, 121, d ≥ 3]] 19
[[127, 118, d
≥ 3]] 19
[[13, 5, d ≥ 3]] 5
—
[[13, 5, d ≥ 3]] 8
—
[[13, 7, d ≥ 3]] 3
—
[[43, 31, d ≥ 3]] 7
—
[[73, 61, d ≥ 3]] 9
—
[[1093, 1079, d ≥ 3]] 3
[[1093, 1072, d
≥ 3]] 3
5.2 BCH Codes—Part II
The material of this subsection is based on the results shown in our paper [104].
We investigate here several properties of q-ary cyclotomic cosets modulo n = q
m
−
1, where q = 2 is a prime power. As applications, several families of nonbinary
Calderbank–Shor–Steane (CSS) quantum codes derived from two distinct Bose–
Chaudhuri–Hocquenghem (BCH) codes are constructed.