Best Known (221−35, 221, s)-Nets in Base 3
(221−35, 221, 1484)-Net over F3 — Constructive and digital
Digital (186, 221, 1484)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (169, 204, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- digital (0, 17, 4)-net over F3, using
(221−35, 221, 9948)-Net over F3 — Digital
Digital (186, 221, 9948)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 9948, F3, 35) (dual of [9948, 9727, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 19732, F3, 35) (dual of [19732, 19511, 36]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3219, 19730, F3, 35) (dual of [19730, 19511, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(3208, 19683, F3, 35) (dual of [19683, 19475, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(311, 47, F3, 5) (dual of [47, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3219, 19730, F3, 35) (dual of [19730, 19511, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 19732, F3, 35) (dual of [19732, 19511, 36]-code), using
(221−35, 221, 5363202)-Net in Base 3 — Upper bound on s
There is no (186, 221, 5363203)-net in base 3, because
- 1 times m-reduction [i] would yield (186, 220, 5363203)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 926 138907 069427 965812 582544 965674 983147 664317 402644 141848 513642 217193 688684 932032 878273 954236 650814 112359 > 3220 [i]