Best Known (67−41, 67, s)-Nets in Base 64
(67−41, 67, 208)-Net over F64 — Constructive and digital
Digital (26, 67, 208)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 23, 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 (3, 44, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64 (see above)
- digital (3, 23, 104)-net over F64, using
(67−41, 67, 288)-Net in Base 64 — Constructive
(26, 67, 288)-net in base 64, using
- t-expansion [i] based on (22, 67, 288)-net in base 64, using
- 24 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- 24 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
(67−41, 67, 425)-Net over F64 — Digital
Digital (26, 67, 425)-net over F64, using
- net from sequence [i] based on digital (26, 424)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 26 and N(F) ≥ 425, using
(67−41, 67, 513)-Net in Base 64
(26, 67, 513)-net in base 64, using
- 5 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
(67−41, 67, 120316)-Net in Base 64 — Upper bound on s
There is no (26, 67, 120317)-net in base 64, because
- 1 times m-reduction [i] would yield (26, 66, 120317)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 161411 682033 939763 126103 067144 205953 603787 613752 275274 643277 322488 021212 762586 148466 672376 863734 469065 557043 154196 132619 > 6466 [i]