Best Known (166−35, 166, s)-Nets in Base 2
(166−35, 166, 260)-Net over F2 — Constructive and digital
Digital (131, 166, 260)-net over F2, using
- 22 times duplication [i] based on digital (129, 164, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 41, 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, 41, 65)-net over F16, using
(166−35, 166, 402)-Net over F2 — Digital
Digital (131, 166, 402)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2166, 402, F2, 2, 35) (dual of [(402, 2), 638, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2166, 512, F2, 2, 35) (dual of [(512, 2), 858, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2166, 1024, F2, 35) (dual of [1024, 858, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- OOA 2-folding [i] based on linear OA(2166, 1024, F2, 35) (dual of [1024, 858, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2166, 512, F2, 2, 35) (dual of [(512, 2), 858, 36]-NRT-code), using
(166−35, 166, 5968)-Net in Base 2 — Upper bound on s
There is no (131, 166, 5969)-net in base 2, because
- 1 times m-reduction [i] would yield (131, 165, 5969)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 46 779741 699277 499551 985283 810701 137744 878226 074202 > 2165 [i]