Best Known (51−14, 51, s)-Nets in Base 16
(51−14, 51, 1542)-Net over F16 — Constructive and digital
Digital (37, 51, 1542)-net over F16, using
- 161 times duplication [i] based on digital (36, 50, 1542)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 8, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 4, 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, 4, 257)-net over F256, using
- digital (7, 14, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 257)-net over F256, using
- digital (14, 28, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- digital (4, 8, 514)-net over F16, using
- generalized (u, u+v)-construction [i] based on
(51−14, 51, 4681)-Net in Base 16 — Constructive
(37, 51, 4681)-net in base 16, using
- 161 times duplication [i] based on (36, 50, 4681)-net in base 16, using
- base change [i] based on digital (26, 40, 4681)-net over F32, using
- net defined by OOA [i] based on linear OOA(3240, 4681, F32, 14, 14) (dual of [(4681, 14), 65494, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3240, 32767, F32, 14) (dual of [32767, 32727, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3240, 32767, F32, 14) (dual of [32767, 32727, 15]-code), using
- net defined by OOA [i] based on linear OOA(3240, 4681, F32, 14, 14) (dual of [(4681, 14), 65494, 15]-NRT-code), using
- base change [i] based on digital (26, 40, 4681)-net over F32, using
(51−14, 51, 20013)-Net over F16 — Digital
Digital (37, 51, 20013)-net over F16, using
(51−14, 51, large)-Net in Base 16 — Upper bound on s
There is no (37, 51, large)-net in base 16, because
- 12 times m-reduction [i] would yield (37, 39, large)-net in base 16, but