Best Known (177, 177+38, s)-Nets in Base 3
(177, 177+38, 896)-Net over F3 — Constructive and digital
Digital (177, 215, 896)-net over F3, using
- t-expansion [i] based on digital (175, 215, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (175, 216, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 54, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 54, 224)-net over F81, using
- 1 times m-reduction [i] based on digital (175, 216, 896)-net over F3, using
(177, 177+38, 4863)-Net over F3 — Digital
Digital (177, 215, 4863)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3215, 4863, F3, 38) (dual of [4863, 4648, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 6614, F3, 38) (dual of [6614, 6399, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [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(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3215, 6614, F3, 38) (dual of [6614, 6399, 39]-code), using
(177, 177+38, 993530)-Net in Base 3 — Upper bound on s
There is no (177, 215, 993531)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 811324 869461 624109 615582 836665 732618 105272 434712 796553 757981 250396 247684 661259 063634 446873 390885 723371 > 3215 [i]