Best Known (186, 186+34, s)-Nets in Base 3
(186, 186+34, 1490)-Net over F3 — Constructive and digital
Digital (186, 220, 1490)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 20, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (166, 200, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 50, 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, 50, 370)-net over F81, using
- digital (3, 20, 10)-net over F3, using
(186, 186+34, 11749)-Net over F3 — Digital
Digital (186, 220, 11749)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3220, 11749, F3, 34) (dual of [11749, 11529, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 19758, F3, 34) (dual of [19758, 19538, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(24) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(321, 75, F3, 8) (dual of [75, 54, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to Ce(33) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3220, 19758, F3, 34) (dual of [19758, 19538, 35]-code), using
(186, 186+34, 5363202)-Net in Base 3 — Upper bound on s
There is no (186, 220, 5363203)-net in base 3, because
- 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]