Best Known (109−24, 109, s)-Nets in Base 2
(109−24, 109, 152)-Net over F2 — Constructive and digital
Digital (85, 109, 152)-net over F2, using
- 21 times duplication [i] based on digital (84, 108, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 27, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 27, 38)-net over F16, using
(109−24, 109, 260)-Net over F2 — Digital
Digital (85, 109, 260)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2109, 260, F2, 2, 24) (dual of [(260, 2), 411, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2109, 520, F2, 24) (dual of [520, 411, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2109, 521, F2, 24) (dual of [521, 412, 25]-code), using
- 1 times truncation [i] based on linear OA(2110, 522, F2, 25) (dual of [522, 412, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2109, 512, F2, 25) (dual of [512, 403, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2100, 512, F2, 23) (dual of [512, 412, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2110, 522, F2, 25) (dual of [522, 412, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2109, 521, F2, 24) (dual of [521, 412, 25]-code), using
- OOA 2-folding [i] based on linear OA(2109, 520, F2, 24) (dual of [520, 411, 25]-code), using
(109−24, 109, 2851)-Net in Base 2 — Upper bound on s
There is no (85, 109, 2852)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 650 505173 481411 588247 951109 697432 > 2109 [i]