Best Known (117, 117+23, s)-Nets in Base 2
(117, 117+23, 390)-Net over F2 — Constructive and digital
Digital (117, 140, 390)-net over F2, using
- 22 times duplication [i] based on digital (115, 138, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 65)-net over F64, using
(117, 117+23, 1147)-Net over F2 — Digital
Digital (117, 140, 1147)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2140, 1147, F2, 3, 23) (dual of [(1147, 3), 3301, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2140, 1375, F2, 3, 23) (dual of [(1375, 3), 3985, 24]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2139, 1375, F2, 3, 23) (dual of [(1375, 3), 3986, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2139, 4125, F2, 23) (dual of [4125, 3986, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2139, 4126, F2, 23) (dual of [4126, 3987, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2133, 4096, F2, 23) (dual of [4096, 3963, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2139, 4126, F2, 23) (dual of [4126, 3987, 24]-code), using
- OOA 3-folding [i] based on linear OA(2139, 4125, F2, 23) (dual of [4125, 3986, 24]-code), using
- 21 times duplication [i] based on linear OOA(2139, 1375, F2, 3, 23) (dual of [(1375, 3), 3986, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2140, 1375, F2, 3, 23) (dual of [(1375, 3), 3985, 24]-NRT-code), using
(117, 117+23, 31240)-Net in Base 2 — Upper bound on s
There is no (117, 140, 31241)-net in base 2, because
- 1 times m-reduction [i] would yield (117, 139, 31241)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 697050 013675 288096 511055 788879 575756 862248 > 2139 [i]