Best Known (58, 58+24, s)-Nets in Base 81
(58, 58+24, 44369)-Net over F81 — Constructive and digital
Digital (58, 82, 44369)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 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 (46, 70, 44287)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 44287, F81, 24, 24) (dual of [(44287, 24), 1062818, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- OA 12-folding and stacking [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
- net defined by OOA [i] based on linear OOA(8170, 44287, F81, 24, 24) (dual of [(44287, 24), 1062818, 25]-NRT-code), using
- digital (0, 12, 82)-net over F81, using
(58, 58+24, 750822)-Net over F81 — Digital
Digital (58, 82, 750822)-net over F81, using
(58, 58+24, large)-Net in Base 81 — Upper bound on s
There is no (58, 82, large)-net in base 81, because
- 22 times m-reduction [i] would yield (58, 60, large)-net in base 81, but