Best Known (180, 180+40, s)-Nets in Base 3
(180, 180+40, 896)-Net over F3 — Constructive and digital
Digital (180, 220, 896)-net over F3, using
- t-expansion [i] based on digital (178, 220, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
(180, 180+40, 4186)-Net over F3 — Digital
Digital (180, 220, 4186)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3220, 4186, F3, 40) (dual of [4186, 3966, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 6602, F3, 40) (dual of [6602, 6382, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 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(311, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3220, 6602, F3, 40) (dual of [6602, 6382, 41]-code), using
(180, 180+40, 735526)-Net in Base 3 — Upper bound on s
There is no (180, 220, 735527)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 926 142810 381595 018933 088665 235004 745724 463448 859238 366831 925289 243113 231973 040990 528452 721791 824493 039961 > 3220 [i]