Best Known (51−27, 51, s)-Nets in Base 64
(51−27, 51, 281)-Net over F64 — Constructive and digital
Digital (24, 51, 281)-net over F64, using
- 1 times m-reduction [i] based on digital (24, 52, 281)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (7, 35, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (3, 17, 104)-net over F64, using
- (u, u+v)-construction [i] based on
(51−27, 51, 337)-Net in Base 64 — Constructive
(24, 51, 337)-net in base 64, using
- 641 times duplication [i] based on (23, 50, 337)-net in base 64, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- (9, 36, 257)-net in base 64, using
- base change [i] based on digital (0, 27, 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
- base change [i] based on digital (0, 27, 257)-net over F256, using
- digital (1, 14, 80)-net over F64, using
- (u, u+v)-construction [i] based on
(51−27, 51, 651)-Net over F64 — Digital
Digital (24, 51, 651)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6451, 651, F64, 27) (dual of [651, 600, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6451, 819, F64, 27) (dual of [819, 768, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 819 | 642−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(6451, 819, F64, 27) (dual of [819, 768, 28]-code), using
(51−27, 51, 795991)-Net in Base 64 — Upper bound on s
There is no (24, 51, 795992)-net in base 64, because
- 1 times m-reduction [i] would yield (24, 50, 795992)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 2 037060 769725 648532 788748 250862 354854 828351 327343 977844 089826 593576 925361 275497 757857 495830 > 6450 [i]