Best Known (81, 81+3, s)-Nets in Base 3
(81, 81+3, large)-Net over F3 — Constructive and digital
Digital (81, 84, large)-net over F3, using
- 3 times m-reduction [i] based on digital (81, 87, large)-net over F3, using
- net defined by OOA [i] based on linear OOA(387, large, F3, 6, 6), using
- appending kth column [i] based on linear OOA(387, large, F3, 5, 6), using
- (u, u+v)-construction [i] based on
- linear OOA(33, 4, F3, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(5;17,3) [i]
- linear OOA(384, 8388602, F3, 5, 6) (dual of [(8388602, 5), 41942926, 7]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(384, large, F3, 3, 6), using
- trace code [i] based on linear OOA(8121, 2796201, F81, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- trace code [i] based on linear OOA(8121, 2796201, F81, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(384, large, F3, 3, 6), using
- linear OOA(33, 4, F3, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(387, large, F3, 5, 6), using
- net defined by OOA [i] based on linear OOA(387, large, F3, 6, 6), using
(81, 81+3, large)-Net in Base 3 — Upper bound on s
There is no (81, 84, large)-net in base 3, because
- 1 times m-reduction [i] would yield (81, 83, large)-net in base 3, but