Best Known (156, 156+41, s)-Nets in Base 3
(156, 156+41, 688)-Net over F3 — Constructive and digital
Digital (156, 197, 688)-net over F3, using
- 31 times duplication [i] based on digital (155, 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
- t-expansion [i] based on digital (154, 196, 688)-net over F3, using
(156, 156+41, 1888)-Net over F3 — Digital
Digital (156, 197, 1888)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3197, 1888, F3, 41) (dual of [1888, 1691, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 2215, F3, 41) (dual of [2215, 2018, 42]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3196, 2214, F3, 41) (dual of [2214, 2018, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3196, 2214, F3, 41) (dual of [2214, 2018, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 2215, F3, 41) (dual of [2215, 2018, 42]-code), using
(156, 156+41, 196798)-Net in Base 3 — Upper bound on s
There is no (156, 197, 196799)-net in base 3, because
- 1 times m-reduction [i] would yield (156, 196, 196799)-net in base 3, but
- 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]