Best Known (111−23, 111, s)-Nets in Base 2
(111−23, 111, 195)-Net over F2 — Constructive and digital
Digital (88, 111, 195)-net over F2, using
- trace code for nets [i] based on digital (14, 37, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(111−23, 111, 341)-Net over F2 — Digital
Digital (88, 111, 341)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2111, 341, F2, 3, 23) (dual of [(341, 3), 912, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- OOA 3-folding [i] based on linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using
(111−23, 111, 5011)-Net in Base 2 — Upper bound on s
There is no (88, 111, 5012)-net in base 2, because
- 1 times m-reduction [i] would yield (88, 110, 5012)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1300 680253 099749 673158 113517 372594 > 2110 [i]