Best Known (57, 57+7, s)-Nets in Base 3
(57, 57+7, 2796204)-Net over F3 — Constructive and digital
Digital (57, 64, 2796204)-net over F3, using
- net defined by OOA [i] based on linear OOA(364, 2796204, F3, 9, 7) (dual of [(2796204, 9), 25165772, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(364, 2796205, F3, 3, 7) (dual of [(2796205, 3), 8388551, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(33, 4, F3, 3, 3) (dual of [(4, 3), 9, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;9,3) [i]
- 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(33, 4, F3, 3, 3) (dual of [(4, 3), 9, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(364, 2796205, F3, 3, 7) (dual of [(2796205, 3), 8388551, 8]-NRT-code), using
(57, 57+7, 4194305)-Net over F3 — Digital
Digital (57, 64, 4194305)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(364, 4194305, F3, 2, 7) (dual of [(4194305, 2), 8388546, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(33, 4, F3, 2, 3) (dual of [(4, 2), 5, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;5,3) [i]
- 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
- linear OOA(33, 4, F3, 2, 3) (dual of [(4, 2), 5, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
(57, 57+7, large)-Net in Base 3 — Upper bound on s
There is no (57, 64, large)-net in base 3, because
- 5 times m-reduction [i] would yield (57, 59, large)-net in base 3, but