Best Known (41−12, 41, s)-Nets in Base 16
(41−12, 41, 1052)-Net over F16 — Constructive and digital
Digital (29, 41, 1052)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 5, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (6, 12, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 257)-net over F256, 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 (see above)
- trace code for nets [i] based on digital (0, 12, 257)-net over F256, using
- digital (1, 5, 24)-net over F16, using
(41−12, 41, 2731)-Net in Base 16 — Constructive
(29, 41, 2731)-net in base 16, using
- net defined by OOA [i] based on OOA(1641, 2731, S16, 12, 12), using
- OA 6-folding and stacking [i] based on OA(1641, 16386, S16, 12), using
- discarding parts of the base [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- OA 6-folding and stacking [i] based on OA(1641, 16386, S16, 12), using
(41−12, 41, 10075)-Net over F16 — Digital
Digital (29, 41, 10075)-net over F16, using
(41−12, 41, large)-Net in Base 16 — Upper bound on s
There is no (29, 41, large)-net in base 16, because
- 10 times m-reduction [i] would yield (29, 31, large)-net in base 16, but