Best Known (165, 165+35, s)-Nets in Base 2
(165, 165+35, 380)-Net over F2 — Constructive and digital
Digital (165, 200, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 40, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
(165, 165+35, 886)-Net over F2 — Digital
Digital (165, 200, 886)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2200, 886, F2, 2, 35) (dual of [(886, 2), 1572, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2200, 1046, F2, 2, 35) (dual of [(1046, 2), 1892, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2200, 2092, F2, 35) (dual of [2092, 1892, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2200, 2093, F2, 35) (dual of [2093, 1893, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2155, 2048, F2, 29) (dual of [2048, 1893, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(212, 45, F2, 5) (dual of [45, 33, 6]-code), using
- discarding factors / shortening the dual code based on 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
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2200, 2093, F2, 35) (dual of [2093, 1893, 36]-code), using
- OOA 2-folding [i] based on linear OA(2200, 2092, F2, 35) (dual of [2092, 1892, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2200, 1046, F2, 2, 35) (dual of [(1046, 2), 1892, 36]-NRT-code), using
(165, 165+35, 23950)-Net in Base 2 — Upper bound on s
There is no (165, 200, 23951)-net in base 2, because
- 1 times m-reduction [i] would yield (165, 199, 23951)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 803770 843821 675393 432065 600556 660817 128713 483482 068522 567232 > 2199 [i]