Best Known (156, 156+40, s)-Nets in Base 3
(156, 156+40, 688)-Net over F3 — Constructive and digital
Digital (156, 196, 688)-net over F3, using
- t-expansion [i] based on digital (154, 196, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 49, 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, 49, 172)-net over F81, using
(156, 156+40, 2074)-Net over F3 — Digital
Digital (156, 196, 2074)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3196, 2074, F3, 40) (dual of [2074, 1878, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 2228, F3, 40) (dual of [2228, 2032, 41]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3193, 2225, F3, 40) (dual of [2225, 2032, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3193, 2225, F3, 40) (dual of [2225, 2032, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 2228, F3, 40) (dual of [2228, 2032, 41]-code), using
(156, 156+40, 196798)-Net in Base 3 — Upper bound on s
There is no (156, 196, 196799)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3279 394116 627382 656620 152861 191093 384287 390773 853064 180970 090269 803298 065636 502747 244115 988441 > 3196 [i]