Best Known (159−29, 159, s)-Nets in Base 3
(159−29, 159, 695)-Net over F3 — Constructive and digital
Digital (130, 159, 695)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 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 (115, 144, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 36, 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, 36, 172)-net over F81, using
- digital (1, 15, 7)-net over F3, using
(159−29, 159, 3358)-Net over F3 — Digital
Digital (130, 159, 3358)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3159, 3358, F3, 29) (dual of [3358, 3199, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 6591, F3, 29) (dual of [6591, 6432, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-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
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3159, 6591, F3, 29) (dual of [6591, 6432, 30]-code), using
(159−29, 159, 732949)-Net in Base 3 — Upper bound on s
There is no (130, 159, 732950)-net in base 3, because
- 1 times m-reduction [i] would yield (130, 158, 732950)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2427 529196 003416 651914 633019 121900 168257 568157 056558 178256 267225 623206 287941 > 3158 [i]