Best Known (86−25, 86, s)-Nets in Base 16
(86−25, 86, 1073)-Net over F16 — Constructive and digital
Digital (61, 86, 1073)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 12, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- digital (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 12, 257)-net over F256, using
- digital (25, 50, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- digital (4, 12, 45)-net over F16, using
(86−25, 86, 1365)-Net in Base 16 — Constructive
(61, 86, 1365)-net in base 16, using
- net defined by OOA [i] based on OOA(1686, 1365, S16, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(1686, 16381, S16, 25), using
- discarding factors based on OA(1686, 16386, S16, 25), using
- discarding parts of the base [i] based on linear OA(12849, 16386, F128, 25) (dual of [16386, 16337, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- discarding parts of the base [i] based on linear OA(12849, 16386, F128, 25) (dual of [16386, 16337, 26]-code), using
- discarding factors based on OA(1686, 16386, S16, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(1686, 16381, S16, 25), using
(86−25, 86, 13503)-Net over F16 — Digital
Digital (61, 86, 13503)-net over F16, using
(86−25, 86, large)-Net in Base 16 — Upper bound on s
There is no (61, 86, large)-net in base 16, because
- 23 times m-reduction [i] would yield (61, 63, large)-net in base 16, but