Best Known (216, 216+12, s)-Nets in Base 3
(216, 216+12, 6990501)-Net over F3 — Constructive and digital
Digital (216, 228, 6990501)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (54, 60, 2796201)-net over F3, using
- net defined by OOA [i] based on linear OOA(360, 2796201, F3, 6, 6) (dual of [(2796201, 6), 16777146, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(360, 2796201, F3, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- appending kth column [i] based on linear OOA(360, 2796201, F3, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(360, 2796201, F3, 6, 6) (dual of [(2796201, 6), 16777146, 7]-NRT-code), using
- 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
- trace code for nets [i] based on digital (44, 56, 1398100)-net over F27, using
- digital (54, 60, 2796201)-net over F3, using
(216, 216+12, large)-Net over F3 — Digital
Digital (216, 228, large)-net over F3, using
- t-expansion [i] based on digital (209, 228, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
(216, 216+12, large)-Net in Base 3 — Upper bound on s
There is no (216, 228, large)-net in base 3, because
- 10 times m-reduction [i] would yield (216, 218, large)-net in base 3, but