Best Known (156, 156+35, s)-Nets in Base 3
(156, 156+35, 702)-Net over F3 — Constructive and digital
Digital (156, 191, 702)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 23, 14)-net over F3, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 6 and N(F) ≥ 14, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
- digital (6, 23, 14)-net over F3, using
(156, 156+35, 3645)-Net over F3 — Digital
Digital (156, 191, 3645)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3191, 3645, F3, 35) (dual of [3645, 3454, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3191, 6591, F3, 35) (dual of [6591, 6400, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-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(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3191, 6591, F3, 35) (dual of [6591, 6400, 36]-code), using
(156, 156+35, 771680)-Net in Base 3 — Upper bound on s
There is no (156, 191, 771681)-net in base 3, because
- 1 times m-reduction [i] would yield (156, 190, 771681)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 498233 503534 683373 854402 624904 570836 154308 517695 538262 031598 240373 714141 145802 118329 947779 > 3190 [i]