Best Known (129, 129+34, s)-Nets in Base 3
(129, 129+34, 640)-Net over F3 — Constructive and digital
Digital (129, 163, 640)-net over F3, using
- t-expansion [i] based on digital (128, 163, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (128, 164, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 41, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 41, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (128, 164, 640)-net over F3, using
(129, 129+34, 1635)-Net over F3 — Digital
Digital (129, 163, 1635)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3163, 1635, F3, 34) (dual of [1635, 1472, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3163, 2216, F3, 34) (dual of [2216, 2053, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(38, 29, F3, 4) (dual of [29, 21, 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(3163, 2216, F3, 34) (dual of [2216, 2053, 35]-code), using
(129, 129+34, 134776)-Net in Base 3 — Upper bound on s
There is no (129, 163, 134777)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 589947 707901 407685 907412 484084 074451 185299 026632 511805 621113 559948 153228 235187 > 3163 [i]