Best Known (117, 117+37, s)-Nets in Base 2
(117, 117+37, 195)-Net over F2 — Constructive and digital
Digital (117, 154, 195)-net over F2, using
- 21 times duplication [i] based on digital (116, 153, 195)-net over F2, using
- trace code for nets [i] based on digital (14, 51, 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
- trace code for nets [i] based on digital (14, 51, 65)-net over F8, using
(117, 117+37, 255)-Net over F2 — Digital
Digital (117, 154, 255)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2154, 255, F2, 2, 37) (dual of [(255, 2), 356, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2154, 256, F2, 2, 37) (dual of [(256, 2), 358, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2154, 512, F2, 37) (dual of [512, 358, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- OOA 2-folding [i] based on linear OA(2154, 512, F2, 37) (dual of [512, 358, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2154, 256, F2, 2, 37) (dual of [(256, 2), 358, 38]-NRT-code), using
(117, 117+37, 2708)-Net in Base 2 — Upper bound on s
There is no (117, 154, 2709)-net in base 2, because
- 1 times m-reduction [i] would yield (117, 153, 2709)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11481 219593 811725 198110 274774 713117 721332 831570 > 2153 [i]