Best Known (178−34, 178, s)-Nets in Base 3
(178−34, 178, 688)-Net over F3 — Constructive and digital
Digital (144, 178, 688)-net over F3, using
- t-expansion [i] based on digital (142, 178, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (142, 180, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 45, 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, 45, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (142, 180, 688)-net over F3, using
(178−34, 178, 3146)-Net over F3 — Digital
Digital (144, 178, 3146)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3178, 3146, F3, 2, 34) (dual of [(3146, 2), 6114, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3178, 3285, F3, 2, 34) (dual of [(3285, 2), 6392, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3178, 6570, F3, 34) (dual of [6570, 6392, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- 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(31, 9, F3, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(3178, 6570, F3, 34) (dual of [6570, 6392, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(3178, 3285, F3, 2, 34) (dual of [(3285, 2), 6392, 35]-NRT-code), using
(178−34, 178, 355332)-Net in Base 3 — Upper bound on s
There is no (144, 178, 355333)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 8 464249 417942 304149 539202 265223 665760 828489 393779 422490 114452 150142 198916 312365 142987 > 3178 [i]