Best Known (219−35, 219, s)-Nets in Base 2
(219−35, 219, 490)-Net over F2 — Constructive and digital
Digital (184, 219, 490)-net over F2, using
- t-expansion [i] based on digital (183, 219, 490)-net over F2, using
- 1 times m-reduction [i] based on digital (183, 220, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 44, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 44, 98)-net over F32, using
- 1 times m-reduction [i] based on digital (183, 220, 490)-net over F2, using
(219−35, 219, 1382)-Net over F2 — Digital
Digital (184, 219, 1382)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2219, 1382, F2, 3, 35) (dual of [(1382, 3), 3927, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2219, 4146, F2, 35) (dual of [4146, 3927, 36]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2217, 4144, F2, 35) (dual of [4144, 3927, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(2205, 4096, F2, 35) (dual of [4096, 3891, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2169, 4096, F2, 29) (dual of [4096, 3927, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2217, 4144, F2, 35) (dual of [4144, 3927, 36]-code), using
- OOA 3-folding [i] based on linear OA(2219, 4146, F2, 35) (dual of [4146, 3927, 36]-code), using
(219−35, 219, 52000)-Net in Base 2 — Upper bound on s
There is no (184, 219, 52001)-net in base 2, because
- 1 times m-reduction [i] would yield (184, 218, 52001)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 421384 088628 326166 244045 578015 302532 052286 890201 192609 702562 500172 > 2218 [i]