Best Known (216−39, 216, s)-Nets in Base 3
(216−39, 216, 896)-Net over F3 — Constructive and digital
Digital (177, 216, 896)-net over F3, using
- t-expansion [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
(216−39, 216, 4303)-Net over F3 — Digital
Digital (177, 216, 4303)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3216, 4303, F3, 39) (dual of [4303, 4087, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 6593, F3, 39) (dual of [6593, 6377, 40]-code), using
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 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(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3216, 6593, F3, 39) (dual of [6593, 6377, 40]-code), using
(216−39, 216, 993530)-Net in Base 3 — Upper bound on s
There is no (177, 216, 993531)-net in base 3, because
- 1 times m-reduction [i] would yield (177, 215, 993531)-net in base 3, but
- 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]