Best Known (185−34, 185, s)-Nets in Base 3
(185−34, 185, 700)-Net over F3 — Constructive and digital
Digital (151, 185, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 21, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (130, 164, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 41, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 41, 172)-net over F81, using
- digital (4, 21, 12)-net over F3, using
(185−34, 185, 3512)-Net over F3 — Digital
Digital (151, 185, 3512)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3185, 3512, F3, 34) (dual of [3512, 3327, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3185, 6593, F3, 34) (dual of [6593, 6408, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 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(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(38, 32, F3, 4) (dual of [32, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3185, 6593, F3, 34) (dual of [6593, 6408, 35]-code), using
(185−34, 185, 558604)-Net in Base 3 — Upper bound on s
There is no (151, 185, 558605)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 18511 491045 267774 224472 653535 575506 994108 575488 072855 542768 872689 993777 060782 985667 450587 > 3185 [i]