Best Known (90, 111, s)-Nets in Base 2
(90, 111, 260)-Net over F2 — Constructive and digital
Digital (90, 111, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (90, 112, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 28, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 28, 65)-net over F16, using
(90, 111, 563)-Net over F2 — Digital
Digital (90, 111, 563)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2111, 563, F2, 3, 21) (dual of [(563, 3), 1578, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2111, 683, F2, 3, 21) (dual of [(683, 3), 1938, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2111, 2049, F2, 21) (dual of [2049, 1938, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2111, 2049, F2, 21) (dual of [2049, 1938, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(2111, 683, F2, 3, 21) (dual of [(683, 3), 1938, 22]-NRT-code), using
(90, 111, 9260)-Net in Base 2 — Upper bound on s
There is no (90, 111, 9261)-net in base 2, because
- 1 times m-reduction [i] would yield (90, 110, 9261)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1298 989131 158622 120558 271547 129878 > 2110 [i]