Best Known (56−16, 56, s)-Nets in Base 16
(56−16, 56, 1148)-Net over F16 — Constructive and digital
Digital (40, 56, 1148)-net over F16, using
- 1 times m-reduction [i] based on digital (40, 57, 1148)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 7, 120)-net over F16, using
- net defined by OOA [i] based on linear OOA(167, 120, F16, 5, 5) (dual of [(120, 5), 593, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- net defined by OOA [i] based on linear OOA(167, 120, F16, 5, 5) (dual of [(120, 5), 593, 6]-NRT-code), using
- digital (8, 16, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 8, 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, 8, 257)-net over F256, using
- digital (17, 34, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 257)-net over F256, using
- digital (2, 7, 120)-net over F16, using
- generalized (u, u+v)-construction [i] based on
(56−16, 56, 2048)-Net in Base 16 — Constructive
(40, 56, 2048)-net in base 16, using
- base change [i] based on digital (16, 32, 2048)-net over F128, using
- 1 times m-reduction [i] based on digital (16, 33, 2048)-net over F128, using
- net defined by OOA [i] based on linear OOA(12833, 2048, F128, 17, 17) (dual of [(2048, 17), 34783, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using
- net defined by OOA [i] based on linear OOA(12833, 2048, F128, 17, 17) (dual of [(2048, 17), 34783, 18]-NRT-code), using
- 1 times m-reduction [i] based on digital (16, 33, 2048)-net over F128, using
(56−16, 56, 13406)-Net over F16 — Digital
Digital (40, 56, 13406)-net over F16, using
(56−16, 56, large)-Net in Base 16 — Upper bound on s
There is no (40, 56, large)-net in base 16, because
- 14 times m-reduction [i] would yield (40, 42, large)-net in base 16, but