Best Known (169, 169+38, s)-Nets in Base 3
(169, 169+38, 896)-Net over F3 — Constructive and digital
Digital (169, 207, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (169, 208, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 52, 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, 52, 224)-net over F81, using
(169, 169+38, 3802)-Net over F3 — Digital
Digital (169, 207, 3802)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3207, 3802, F3, 38) (dual of [3802, 3595, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3207, 6591, F3, 38) (dual of [6591, 6384, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [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(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, 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(37) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3207, 6591, F3, 38) (dual of [6591, 6384, 39]-code), using
(169, 169+38, 625580)-Net in Base 3 — Upper bound on s
There is no (169, 207, 625581)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 580 907200 115302 717246 425780 445914 141199 340765 531372 147227 319209 942918 237607 674759 805651 767997 795211 > 3207 [i]