Best Known (168−12, 168, s)-Nets in Base 3
(168−12, 168, 4194300)-Net over F3 — Constructive and digital
Digital (156, 168, 4194300)-net over F3, using
- trace code for nets [i] based on digital (44, 56, 1398100)-net over F27, using
- net defined by OOA [i] based on linear OOA(2756, 1398100, F27, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2756, 8388600, F27, 12) (dual of [8388600, 8388544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2756, 8388600, F27, 12) (dual of [8388600, 8388544, 13]-code), using
- net defined by OOA [i] based on linear OOA(2756, 1398100, F27, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
(168−12, 168, large)-Net over F3 — Digital
Digital (156, 168, large)-net over F3, using
- 33 times duplication [i] based on digital (153, 165, large)-net over F3, using
- t-expansion [i] based on digital (151, 165, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- t-expansion [i] based on digital (151, 165, large)-net over F3, using
(168−12, 168, large)-Net in Base 3 — Upper bound on s
There is no (156, 168, large)-net in base 3, because
- 10 times m-reduction [i] would yield (156, 158, large)-net in base 3, but