Best Known (55, 78, s)-Nets in Base 81
(55, 78, 48395)-Net over F81 — Constructive and digital
Digital (55, 78, 48395)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (44, 67, 48313)-net over F81, using
- net defined by OOA [i] based on linear OOA(8167, 48313, F81, 23, 23) (dual of [(48313, 23), 1111132, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 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(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- net defined by OOA [i] based on linear OOA(8167, 48313, F81, 23, 23) (dual of [(48313, 23), 1111132, 24]-NRT-code), using
- digital (0, 11, 82)-net over F81, using
(55, 78, 661024)-Net over F81 — Digital
Digital (55, 78, 661024)-net over F81, using
(55, 78, large)-Net in Base 81 — Upper bound on s
There is no (55, 78, large)-net in base 81, because
- 21 times m-reduction [i] would yield (55, 57, large)-net in base 81, but