Best Known (165−27, 165, s)-Nets in Base 2
(165−27, 165, 390)-Net over F2 — Constructive and digital
Digital (138, 165, 390)-net over F2, using
- 23 times duplication [i] based on digital (135, 162, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 65)-net over F64, using
(165−27, 165, 1211)-Net over F2 — Digital
Digital (138, 165, 1211)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2165, 1211, F2, 3, 27) (dual of [(1211, 3), 3468, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2165, 1376, F2, 3, 27) (dual of [(1376, 3), 3963, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2165, 4128, F2, 27) (dual of [4128, 3963, 28]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2163, 4126, F2, 27) (dual of [4126, 3963, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- 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(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(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(26) ⊂ Ce(22) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2163, 4126, F2, 27) (dual of [4126, 3963, 28]-code), using
- OOA 3-folding [i] based on linear OA(2165, 4128, F2, 27) (dual of [4128, 3963, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(2165, 1376, F2, 3, 27) (dual of [(1376, 3), 3963, 28]-NRT-code), using
(165−27, 165, 35545)-Net in Base 2 — Upper bound on s
There is no (138, 165, 35546)-net in base 2, because
- 1 times m-reduction [i] would yield (138, 164, 35546)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 23 389208 518447 797715 264844 651692 544594 895506 818788 > 2164 [i]