Best Known (81, 81+20, s)-Nets in Base 16
(81, 81+20, 104875)-Net over F16 — Constructive and digital
Digital (81, 101, 104875)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (71, 91, 104858)-net over F16, using
- net defined by OOA [i] based on linear OOA(1691, 104858, F16, 20, 20) (dual of [(104858, 20), 2097069, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1691, 1048580, F16, 20) (dual of [1048580, 1048489, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1691, 1048581, F16, 20) (dual of [1048581, 1048490, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(1691, 1048581, F16, 20) (dual of [1048581, 1048490, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(1691, 1048580, F16, 20) (dual of [1048580, 1048489, 21]-code), using
- net defined by OOA [i] based on linear OOA(1691, 104858, F16, 20, 20) (dual of [(104858, 20), 2097069, 21]-NRT-code), using
- digital (0, 10, 17)-net over F16, using
(81, 81+20, 1330380)-Net over F16 — Digital
Digital (81, 101, 1330380)-net over F16, using
(81, 81+20, large)-Net in Base 16 — Upper bound on s
There is no (81, 101, large)-net in base 16, because
- 18 times m-reduction [i] would yield (81, 83, large)-net in base 16, but