Best Known (56, 56+23, s)-Nets in Base 16
(56, 56+23, 1073)-Net over F16 — Constructive and digital
Digital (56, 79, 1073)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 11, 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 (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 257)-net over F256, using
- digital (23, 46, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 257)-net over F256, using
- digital (4, 11, 45)-net over F16, using
(56, 56+23, 1489)-Net in Base 16 — Constructive
(56, 79, 1489)-net in base 16, using
- net defined by OOA [i] based on OOA(1679, 1489, S16, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(1679, 16380, S16, 23), using
- discarding factors based on OA(1679, 16386, S16, 23), using
- discarding parts of the base [i] based on linear OA(12845, 16386, F128, 23) (dual of [16386, 16341, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12843, 16384, F128, 22) (dual of [16384, 16341, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(12845, 16386, F128, 23) (dual of [16386, 16341, 24]-code), using
- discarding factors based on OA(1679, 16386, S16, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(1679, 16380, S16, 23), using
(56, 56+23, 12736)-Net over F16 — Digital
Digital (56, 79, 12736)-net over F16, using
(56, 56+23, large)-Net in Base 16 — Upper bound on s
There is no (56, 79, large)-net in base 16, because
- 21 times m-reduction [i] would yield (56, 58, large)-net in base 16, but