Best Known (110−22, 110, s)-Nets in Base 2
(110−22, 110, 196)-Net over F2 — Constructive and digital
Digital (88, 110, 196)-net over F2, using
- 22 times duplication [i] based on digital (86, 108, 196)-net over F2, using
- trace code for nets [i] based on digital (5, 27, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 27, 49)-net over F16, using
(110−22, 110, 381)-Net over F2 — Digital
Digital (88, 110, 381)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2110, 381, F2, 2, 22) (dual of [(381, 2), 652, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2110, 511, F2, 2, 22) (dual of [(511, 2), 912, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2110, 1022, F2, 22) (dual of [1022, 912, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2110, 1023, F2, 22) (dual of [1023, 913, 23]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(2110, 1023, F2, 22) (dual of [1023, 913, 23]-code), using
- OOA 2-folding [i] based on linear OA(2110, 1022, F2, 22) (dual of [1022, 912, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(2110, 511, F2, 2, 22) (dual of [(511, 2), 912, 23]-NRT-code), using
(110−22, 110, 5011)-Net in Base 2 — Upper bound on s
There is no (88, 110, 5012)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1300 680253 099749 673158 113517 372594 > 2110 [i]