Best Known (158, 158+36, s)-Nets in Base 3
(158, 158+36, 700)-Net over F3 — Constructive and digital
Digital (158, 194, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 22, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (136, 172, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 43, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 43, 172)-net over F81, using
- digital (4, 22, 12)-net over F3, using
(158, 158+36, 3426)-Net over F3 — Digital
Digital (158, 194, 3426)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3194, 3426, F3, 36) (dual of [3426, 3232, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3194, 6578, F3, 36) (dual of [6578, 6384, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3194, 6578, F3, 36) (dual of [6578, 6384, 37]-code), using
(158, 158+36, 524073)-Net in Base 3 — Upper bound on s
There is no (158, 194, 524074)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 364 356891 708996 164896 956500 862282 380059 009795 194786 732101 095457 705415 746774 723052 396453 547365 > 3194 [i]