Best Known (136−12, 136, s)-Nets in Base 3
(136−12, 136, 1398146)-Net over F3 — Constructive and digital
Digital (124, 136, 1398146)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (10, 16, 46)-net over F3, using
- net defined by OOA [i] based on linear OOA(316, 46, F3, 6, 6) (dual of [(46, 6), 260, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(316, 46, F3, 5, 6) (dual of [(46, 5), 214, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(33, 4, F3, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(5;17,3) [i]
- linear OOA(313, 42, F3, 5, 6) (dual of [(42, 5), 197, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (7, 13, 42)-net over F3, using
- linear OOA(33, 4, F3, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(316, 46, F3, 5, 6) (dual of [(46, 5), 214, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(316, 46, F3, 6, 6) (dual of [(46, 6), 260, 7]-NRT-code), using
- digital (108, 120, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3120, 1398100, F3, 12, 12) (dual of [(1398100, 12), 16777080, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3120, 8388600, F3, 12) (dual of [8388600, 8388480, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3120, 8388600, F3, 12) (dual of [8388600, 8388480, 13]-code), using
- net defined by OOA [i] based on linear OOA(3120, 1398100, F3, 12, 12) (dual of [(1398100, 12), 16777080, 13]-NRT-code), using
- digital (10, 16, 46)-net over F3, using
(136−12, 136, 6252916)-Net over F3 — Digital
Digital (124, 136, 6252916)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3136, 6252916, F3, 12) (dual of [6252916, 6252780, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, large, F3, 12) (dual of [large, large−136, 13]-code), using
- strength reduction [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]
- strength reduction [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, large, F3, 12) (dual of [large, large−136, 13]-code), using
(136−12, 136, large)-Net in Base 3 — Upper bound on s
There is no (124, 136, large)-net in base 3, because
- 10 times m-reduction [i] would yield (124, 126, large)-net in base 3, but