Best Known (187, 187+35, s)-Nets in Base 2
(187, 187+35, 520)-Net over F2 — Constructive and digital
Digital (187, 222, 520)-net over F2, using
- 22 times duplication [i] based on digital (185, 220, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 44, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 44, 104)-net over F32, using
(187, 187+35, 1811)-Net over F2 — Digital
Digital (187, 222, 1811)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2222, 1811, F2, 4, 35) (dual of [(1811, 4), 7022, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2222, 2048, F2, 4, 35) (dual of [(2048, 4), 7970, 36]-NRT-code), using
- OOA 4-folding [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]
- OOA 4-folding [i] based on linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2222, 2048, F2, 4, 35) (dual of [(2048, 4), 7970, 36]-NRT-code), using
(187, 187+35, 58769)-Net in Base 2 — Upper bound on s
There is no (187, 222, 58770)-net in base 2, because
- 1 times m-reduction [i] would yield (187, 221, 58770)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 370680 498535 854746 366259 460725 528113 637081 816276 825899 428831 498203 > 2221 [i]