Best Known (171, 171+38, s)-Nets in Base 3
(171, 171+38, 896)-Net over F3 — Constructive and digital
Digital (171, 209, 896)-net over F3, using
- 31 times duplication [i] based on digital (170, 208, 896)-net over F3, using
- t-expansion [i] based on digital (169, 208, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 52, 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, 52, 224)-net over F81, using
- t-expansion [i] based on digital (169, 208, 896)-net over F3, using
(171, 171+38, 4043)-Net over F3 — Digital
Digital (171, 209, 4043)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3209, 4043, F3, 38) (dual of [4043, 3834, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3209, 6594, F3, 38) (dual of [6594, 6385, 39]-code), using
- construction XX applied to Ce(37) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [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(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(37) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3209, 6594, F3, 38) (dual of [6594, 6385, 39]-code), using
(171, 171+38, 702276)-Net in Base 3 — Upper bound on s
There is no (171, 209, 702277)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5228 211180 483822 644205 992035 939119 304670 434471 382172 838799 849822 069378 222041 752344 191819 518041 663019 > 3209 [i]