Best Known (168−26, 168, s)-Nets in Base 3
(168−26, 168, 1521)-Net over F3 — Constructive and digital
Digital (142, 168, 1521)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-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 (1, 14, 7)-net over F3, using
(168−26, 168, 10217)-Net over F3 — Digital
Digital (142, 168, 10217)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3168, 10217, F3, 26) (dual of [10217, 10049, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3168, 19736, F3, 26) (dual of [19736, 19568, 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(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3168, 19736, F3, 26) (dual of [19736, 19568, 27]-code), using
(168−26, 168, 4151925)-Net in Base 3 — Upper bound on s
There is no (142, 168, 4151926)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 143 341347 479415 596290 970269 810287 639940 988946 170237 326313 814044 928768 903824 485173 > 3168 [i]