Best Known (45−23, 45, s)-Nets in Base 128
(45−23, 45, 1489)-Net over F128 — Constructive and digital
Digital (22, 45, 1489)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
(45−23, 45, 3742)-Net over F128 — Digital
Digital (22, 45, 3742)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12845, 3742, F128, 4, 23) (dual of [(3742, 4), 14923, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12845, 4096, F128, 4, 23) (dual of [(4096, 4), 16339, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 4-folding [i] based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(12845, 4096, F128, 4, 23) (dual of [(4096, 4), 16339, 24]-NRT-code), using
(45−23, 45, large)-Net in Base 128 — Upper bound on s
There is no (22, 45, large)-net in base 128, because
- 21 times m-reduction [i] would yield (22, 24, large)-net in base 128, but