Best Known (54, 54+21, s)-Nets in Base 81
(54, 54+21, 53294)-Net over F81 — Constructive and digital
Digital (54, 75, 53294)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (40, 61, 53144)-net over F81, using
- net defined by OOA [i] based on linear OOA(8161, 53144, F81, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using
- net defined by OOA [i] based on linear OOA(8161, 53144, F81, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
- digital (4, 14, 150)-net over F81, using
(54, 54+21, 1489491)-Net over F81 — Digital
Digital (54, 75, 1489491)-net over F81, using
(54, 54+21, large)-Net in Base 81 — Upper bound on s
There is no (54, 75, large)-net in base 81, because
- 19 times m-reduction [i] would yield (54, 56, large)-net in base 81, but