Best Known (59−33, 59, s)-Nets in Base 64
(59−33, 59, 281)-Net over F64 — Constructive and digital
Digital (26, 59, 281)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 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, 40, 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, 19, 104)-net over F64, using
(59−33, 59, 306)-Net in Base 64 — Constructive
(26, 59, 306)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 19, 129)-net in base 64, using
- 2 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- 2 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- digital (7, 40, 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
- (3, 19, 129)-net in base 64, using
(59−33, 59, 451)-Net over F64 — Digital
Digital (26, 59, 451)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6459, 451, F64, 33) (dual of [451, 392, 34]-code), using
- 42 step Varšamov–Edel lengthening with (ri) = (1, 41 times 0) [i] based on linear OA(6458, 408, F64, 33) (dual of [408, 350, 34]-code), using
- extended algebraic-geometric code AGe(F,374P) [i] based on function field F/F64 with g(F) = 25 and N(F) ≥ 408, using
- 42 step Varšamov–Edel lengthening with (ri) = (1, 41 times 0) [i] based on linear OA(6458, 408, F64, 33) (dual of [408, 350, 34]-code), using
(59−33, 59, 513)-Net in Base 64
(26, 59, 513)-net in base 64, using
- 13 times m-reduction [i] based on (26, 72, 513)-net in base 64, using
- base change [i] based on digital (8, 54, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- base change [i] based on digital (8, 54, 513)-net over F256, using
(59−33, 59, 380707)-Net in Base 64 — Upper bound on s
There is no (26, 59, 380708)-net in base 64, because
- 1 times m-reduction [i] would yield (26, 58, 380708)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 573 380301 466470 289402 762710 241707 156239 550017 505411 642329 876812 920162 457621 192392 722124 858701 845881 606790 > 6458 [i]