Best Known (57−17, 57, s)-Nets in Base 81
(57−17, 57, 66512)-Net over F81 — Constructive and digital
Digital (40, 57, 66512)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (32, 49, 66430)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 66430, F81, 17, 17) (dual of [(66430, 17), 1129261, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using
- net defined by OOA [i] based on linear OOA(8149, 66430, F81, 17, 17) (dual of [(66430, 17), 1129261, 18]-NRT-code), using
- digital (0, 8, 82)-net over F81, using
(57−17, 57, 535098)-Net over F81 — Digital
Digital (40, 57, 535098)-net over F81, using
(57−17, 57, large)-Net in Base 81 — Upper bound on s
There is no (40, 57, large)-net in base 81, because
- 15 times m-reduction [i] would yield (40, 42, large)-net in base 81, but