Best Known (133, 133+28, s)-Nets in Base 3
(133, 133+28, 704)-Net over F3 — Constructive and digital
Digital (133, 161, 704)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 16)-net over F3, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 7 and N(F) ≥ 16, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 172)-net over F81, using
- digital (7, 21, 16)-net over F3, using
(133, 133+28, 4529)-Net over F3 — Digital
Digital (133, 161, 4529)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3161, 4529, F3, 28) (dual of [4529, 4368, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 6617, F3, 28) (dual of [6617, 6456, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(316, 56, F3, 7) (dual of [56, 40, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3161, 6617, F3, 28) (dual of [6617, 6456, 29]-code), using
(133, 133+28, 927504)-Net in Base 3 — Upper bound on s
There is no (133, 161, 927505)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 65543 211343 617923 207642 430041 678730 283535 505369 179076 496409 279379 872546 626329 > 3161 [i]