Best Known (169−26, 169, s)-Nets in Base 3
(169−26, 169, 1522)-Net over F3 — Constructive and digital
Digital (143, 169, 1522)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (128, 154, 1514)-net over F3, using
- net defined by OOA [i] based on linear OOA(3154, 1514, F3, 26, 26) (dual of [(1514, 26), 39210, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3154, 19682, F3, 26) (dual of [19682, 19528, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3154, 19682, F3, 26) (dual of [19682, 19528, 27]-code), using
- net defined by OOA [i] based on linear OOA(3154, 1514, F3, 26, 26) (dual of [(1514, 26), 39210, 27]-NRT-code), using
- digital (2, 15, 8)-net over F3, using
(169−26, 169, 10697)-Net over F3 — Digital
Digital (143, 169, 10697)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3169, 10697, F3, 26) (dual of [10697, 10528, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 19743, F3, 26) (dual of [19743, 19574, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(315, 60, F3, 6) (dual of [60, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3169, 19743, F3, 26) (dual of [19743, 19574, 27]-code), using
(169−26, 169, 4518052)-Net in Base 3 — Upper bound on s
There is no (143, 169, 4518053)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 430 023731 066867 539822 796584 415653 222601 086609 877020 655717 453040 572353 253507 599419 > 3169 [i]