Best Known (41−12, 41, s)-Nets in Base 81
(41−12, 41, 88674)-Net over F81 — Constructive and digital
Digital (29, 41, 88674)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (22, 34, 88574)-net over F81, using
- net defined by OOA [i] based on linear OOA(8134, 88574, F81, 12, 12) (dual of [(88574, 12), 1062854, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- net defined by OOA [i] based on linear OOA(8134, 88574, F81, 12, 12) (dual of [(88574, 12), 1062854, 13]-NRT-code), using
- digital (1, 7, 100)-net over F81, using
(41−12, 41, 796840)-Net over F81 — Digital
Digital (29, 41, 796840)-net over F81, using
(41−12, 41, large)-Net in Base 81 — Upper bound on s
There is no (29, 41, large)-net in base 81, because
- 10 times m-reduction [i] would yield (29, 31, large)-net in base 81, but