Best Known (76, 76+7, s)-Nets in Base 3
(76, 76+7, 5592405)-Net over F3 — Constructive and digital
Digital (76, 83, 5592405)-net over F3, using
- net defined by OOA [i] based on linear OOA(383, 5592405, F3, 9, 7) (dual of [(5592405, 9), 50331562, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(383, 5592406, F3, 3, 7) (dual of [(5592406, 3), 16777135, 8]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(32, 4, F3, 3, 2) (dual of [(4, 3), 10, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;10,3) [i]
- linear OOA(320, 2796201, F3, 3, 3) (dual of [(2796201, 3), 8388583, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(320, 3713023, F3, 3, 3) (dual of [(3713023, 3), 11139049, 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(32, 4, F3, 3, 2) (dual of [(4, 3), 10, 3]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(383, 5592406, F3, 3, 7) (dual of [(5592406, 3), 16777135, 8]-NRT-code), using
(76, 76+7, large)-Net over F3 — Digital
Digital (76, 83, large)-net over F3, using
- 310 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
(76, 76+7, large)-Net in Base 3 — Upper bound on s
There is no (76, 83, large)-net in base 3, because
- 5 times m-reduction [i] would yield (76, 78, large)-net in base 3, but