Best Known (187−31, 187, s)-Nets in Base 2
(187−31, 187, 390)-Net over F2 — Constructive and digital
Digital (156, 187, 390)-net over F2, using
- 21 times duplication [i] based on digital (155, 186, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 31, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 31, 65)-net over F64, using
(187−31, 187, 1191)-Net over F2 — Digital
Digital (156, 187, 1191)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2187, 1191, F2, 3, 31) (dual of [(1191, 3), 3386, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2187, 1375, F2, 3, 31) (dual of [(1375, 3), 3938, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2187, 4125, F2, 31) (dual of [4125, 3938, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2187, 4126, F2, 31) (dual of [4126, 3939, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2181, 4096, F2, 31) (dual of [4096, 3915, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on 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(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2187, 4126, F2, 31) (dual of [4126, 3939, 32]-code), using
- OOA 3-folding [i] based on linear OA(2187, 4125, F2, 31) (dual of [4125, 3938, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2187, 1375, F2, 3, 31) (dual of [(1375, 3), 3938, 32]-NRT-code), using
(187−31, 187, 34694)-Net in Base 2 — Upper bound on s
There is no (156, 187, 34695)-net in base 2, because
- 1 times m-reduction [i] would yield (156, 186, 34695)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 98 091651 248257 116758 354084 363128 356097 853242 761864 225568 > 2186 [i]