Best Known (72, 72+7, s)-Nets in Base 3
(72, 72+7, 3541672)-Net over F3 — Constructive and digital
Digital (72, 79, 3541672)-net over F3, using
- net defined by OOA [i] based on linear OOA(379, 3541672, F3, 9, 7) (dual of [(3541672, 9), 31874969, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(379, 3541673, F3, 3, 7) (dual of [(3541673, 3), 10624940, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(318, 745472, F3, 3, 3) (dual of [(745472, 3), 2236398, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(318, 745472, F3, 2, 3) (dual of [(745472, 2), 1490926, 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(318, 745472, F3, 3, 3) (dual of [(745472, 3), 2236398, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(379, 3541673, F3, 3, 7) (dual of [(3541673, 3), 10624940, 8]-NRT-code), using
(72, 72+7, large)-Net over F3 — Digital
Digital (72, 79, large)-net over F3, using
- 36 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
(72, 72+7, large)-Net in Base 3 — Upper bound on s
There is no (72, 79, large)-net in base 3, because
- 5 times m-reduction [i] would yield (72, 74, large)-net in base 3, but