Best Known (28−6, 28, s)-Nets in Base 81
(28−6, 28, 5592402)-Net over F81 — Constructive and digital
Digital (22, 28, 5592402)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 7, large)-net over F81, using
- net defined by OOA [i] based on linear OOA(817, large, F81, 3, 3), using
- appending kth column [i] based on linear OOA(817, large, F81, 2, 3), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(817, large, F81, 3) (dual of [large, large−7, 4]-code), using
- appending kth column [i] based on linear OOA(817, large, F81, 2, 3), using
- net defined by OOA [i] based on linear OOA(817, large, F81, 3, 3), using
- digital (15, 21, 2796201)-net over F81, using
- net defined by OOA [i] based on linear OOA(8121, 2796201, F81, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- OA 3-folding and stacking [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]
- OA 3-folding and stacking [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- net defined by OOA [i] based on linear OOA(8121, 2796201, F81, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- digital (4, 7, large)-net over F81, using
(28−6, 28, large)-Net over F81 — Digital
Digital (22, 28, large)-net over F81, using
- t-expansion [i] based on digital (21, 28, large)-net over F81, using
- 1 times m-reduction [i] based on digital (21, 29, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- 1 times m-reduction [i] based on digital (21, 29, large)-net over F81, using
(28−6, 28, large)-Net in Base 81 — Upper bound on s
There is no (22, 28, large)-net in base 81, because
- 4 times m-reduction [i] would yield (22, 24, large)-net in base 81, but