Best Known (80−7, 80, s)-Nets in Base 3
(80−7, 80, 4287144)-Net over F3 — Constructive and digital
Digital (73, 80, 4287144)-net over F3, using
- net defined by OOA [i] based on linear OOA(380, 4287144, F3, 9, 7) (dual of [(4287144, 9), 38584216, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(380, 4287145, F3, 3, 7) (dual of [(4287145, 3), 12861355, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(319, 1490944, F3, 3, 3) (dual of [(1490944, 3), 4472813, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(319, 1490944, F3, 2, 3) (dual of [(1490944, 2), 2981869, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(319, 1490944, F3, 3) (dual of [1490944, 1490925, 4]-code or 1490944-cap in PG(18,3)), using
- appending kth column [i] based on linear OOA(319, 1490944, F3, 2, 3) (dual of [(1490944, 2), 2981869, 4]-NRT-code), using
- linear OOA(361, 2796201, F3, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- linear OOA(319, 1490944, F3, 3, 3) (dual of [(1490944, 3), 4472813, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(380, 4287145, F3, 3, 7) (dual of [(4287145, 3), 12861355, 8]-NRT-code), using
(80−7, 80, large)-Net over F3 — Digital
Digital (73, 80, large)-net over F3, using
- 37 times duplication [i] based on digital (66, 73, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(373, large, F3, 7) (dual of [large, large−73, 8]-code), using
- 12 times code embedding in larger space [i] 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]
- 12 times code embedding in larger space [i] based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(373, large, F3, 7) (dual of [large, large−73, 8]-code), using
(80−7, 80, large)-Net in Base 3 — Upper bound on s
There is no (73, 80, large)-net in base 3, because
- 5 times m-reduction [i] would yield (73, 75, large)-net in base 3, but