Best Known (72, 72+18, s)-Nets in Base 2
(72, 72+18, 180)-Net over F2 — Constructive and digital
Digital (72, 90, 180)-net over F2, using
- 22 times duplication [i] based on digital (70, 88, 180)-net over F2, using
- trace code for nets [i] based on digital (4, 22, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- trace code for nets [i] based on digital (4, 22, 45)-net over F16, using
(72, 72+18, 354)-Net over F2 — Digital
Digital (72, 90, 354)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(290, 354, F2, 2, 18) (dual of [(354, 2), 618, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(290, 511, F2, 2, 18) (dual of [(511, 2), 932, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(290, 1022, F2, 18) (dual of [1022, 932, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(290, 1023, F2, 18) (dual of [1023, 933, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(290, 1023, F2, 18) (dual of [1023, 933, 19]-code), using
- OOA 2-folding [i] based on linear OA(290, 1022, F2, 18) (dual of [1022, 932, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(290, 511, F2, 2, 18) (dual of [(511, 2), 932, 19]-NRT-code), using
(72, 72+18, 4233)-Net in Base 2 — Upper bound on s
There is no (72, 90, 4234)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1238 685394 290270 790534 217011 > 290 [i]