Best Known (169−34, 169, s)-Nets in Base 3
(169−34, 169, 688)-Net over F3 — Constructive and digital
Digital (135, 169, 688)-net over F3, using
- 31 times duplication [i] based on digital (134, 168, 688)-net over F3, using
- t-expansion [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
- t-expansion [i] based on digital (133, 168, 688)-net over F3, using
(169−34, 169, 2015)-Net over F3 — Digital
Digital (135, 169, 2015)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3169, 2015, F3, 34) (dual of [2015, 1846, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 2229, F3, 34) (dual of [2229, 2060, 35]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3165, 2225, F3, 34) (dual of [2225, 2060, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [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(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(3165, 2225, F3, 34) (dual of [2225, 2060, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 2229, F3, 34) (dual of [2229, 2060, 35]-code), using
(169−34, 169, 198621)-Net in Base 3 — Upper bound on s
There is no (135, 169, 198622)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 430 050690 406718 544759 574045 551642 831076 392513 633271 623549 610573 063170 707543 370141 > 3169 [i]