Best Known (56, 56+7, s)-Nets in Base 3
(56, 56+7, 2796200)-Net over F3 — Constructive and digital
Digital (56, 63, 2796200)-net over F3, using
- 32 times duplication [i] based on digital (54, 61, 2796200)-net over F3, using
- net defined by OOA [i] based on linear OOA(361, 2796200, F3, 7, 7) (dual of [(2796200, 7), 19573339, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(361, 8388601, F3, 7) (dual of [8388601, 8388540, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(361, 8388601, F3, 7) (dual of [8388601, 8388540, 8]-code), using
- net defined by OOA [i] based on linear OOA(361, 2796200, F3, 7, 7) (dual of [(2796200, 7), 19573339, 8]-NRT-code), using
(56, 56+7, 4194302)-Net over F3 — Digital
Digital (56, 63, 4194302)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(363, 4194302, F3, 2, 7) (dual of [(4194302, 2), 8388541, 8]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(361, 4194301, F3, 2, 7) (dual of [(4194301, 2), 8388541, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(361, 8388602, F3, 7) (dual of [8388602, 8388541, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- OOA 2-folding [i] based on linear OA(361, 8388602, F3, 7) (dual of [8388602, 8388541, 8]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(361, 4194301, F3, 2, 7) (dual of [(4194301, 2), 8388541, 8]-NRT-code), using
(56, 56+7, large)-Net in Base 3 — Upper bound on s
There is no (56, 63, large)-net in base 3, because
- 5 times m-reduction [i] would yield (56, 58, large)-net in base 3, but