Best Known (227−34, 227, s)-Nets in Base 2
(227−34, 227, 624)-Net over F2 — Constructive and digital
Digital (193, 227, 624)-net over F2, using
- 1 times m-reduction [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
(227−34, 227, 2087)-Net over F2 — Digital
Digital (193, 227, 2087)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 2087, F2, 3, 34) (dual of [(2087, 3), 6034, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 2741, F2, 3, 34) (dual of [(2741, 3), 7996, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2227, 8223, F2, 34) (dual of [8223, 7996, 35]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2228, 8224, F2, 35) (dual of [8224, 7996, 36]-code), using
- OOA 3-folding [i] based on linear OA(2227, 8223, F2, 34) (dual of [8223, 7996, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 2741, F2, 3, 34) (dual of [(2741, 3), 7996, 35]-NRT-code), using
(227−34, 227, 75064)-Net in Base 2 — Upper bound on s
There is no (193, 227, 75065)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 215 688448 574899 667076 280232 898264 847983 279950 171495 978736 580241 359418 > 2227 [i]