Best Known (230−35, 230, s)-Nets in Base 2
(230−35, 230, 624)-Net over F2 — Constructive and digital
Digital (195, 230, 624)-net over F2, using
- 22 times duplication [i] based on digital (193, 228, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 38, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 38, 104)-net over F64, using
(230−35, 230, 2056)-Net over F2 — Digital
Digital (195, 230, 2056)-net over F2, using
- 22 times duplication [i] based on digital (193, 228, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 2056, F2, 4, 35) (dual of [(2056, 4), 7996, 36]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2228, 8224, F2, 35) (dual of [8224, 7996, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- OOA 4-folding [i] based on linear OA(2228, 8224, F2, 35) (dual of [8224, 7996, 36]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 2056, F2, 4, 35) (dual of [(2056, 4), 7996, 36]-NRT-code), using
(230−35, 230, 81444)-Net in Base 2 — Upper bound on s
There is no (195, 230, 81445)-net in base 2, because
- 1 times m-reduction [i] would yield (195, 229, 81445)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 862 764657 546725 510577 497092 473162 146023 847533 116136 458106 098903 553908 > 2229 [i]