Best Known (78, 78+7, s)-Nets in Base 3
(78, 78+7, 5592441)-Net over F3 — Constructive and digital
Digital (78, 85, 5592441)-net over F3, using
- net defined by OOA [i] based on linear OOA(385, 5592441, F3, 9, 7) (dual of [(5592441, 9), 50331884, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(385, 5592442, F3, 3, 7) (dual of [(5592442, 3), 16777241, 8]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(34, 40, F3, 3, 2) (dual of [(40, 3), 116, 3]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(34, 40, F3, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
- appending kth column [i] based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- appending kth column [i] based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- appending 1 arbitrary column [i] based on linear OOA(34, 40, F3, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
- 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(34, 40, F3, 3, 2) (dual of [(40, 3), 116, 3]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(385, 5592442, F3, 3, 7) (dual of [(5592442, 3), 16777241, 8]-NRT-code), using
(78, 78+7, large)-Net over F3 — Digital
Digital (78, 85, large)-net over F3, using
- 1 times m-reduction [i] based on digital (78, 86, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(386, large, F3, 8) (dual of [large, large−86, 9]-code), using
- 10 times code embedding in larger space [i] based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 10 times code embedding in larger space [i] based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(386, large, F3, 8) (dual of [large, large−86, 9]-code), using
(78, 78+7, large)-Net in Base 3 — Upper bound on s
There is no (78, 85, large)-net in base 3, because
- 5 times m-reduction [i] would yield (78, 80, large)-net in base 3, but