Best Known (61−37, 61, s)-Nets in Base 64
(61−37, 61, 208)-Net over F64 — Constructive and digital
Digital (24, 61, 208)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 21, 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, 40, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64 (see above)
- digital (3, 21, 104)-net over F64, using
(61−37, 61, 288)-Net in Base 64 — Constructive
(24, 61, 288)-net in base 64, using
- t-expansion [i] based on (22, 61, 288)-net in base 64, using
- 30 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
- 30 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
(61−37, 61, 342)-Net over F64 — Digital
Digital (24, 61, 342)-net over F64, using
- t-expansion [i] based on digital (20, 61, 342)-net over F64, using
- net from sequence [i] based on digital (20, 341)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 20 and N(F) ≥ 342, using
- net from sequence [i] based on digital (20, 341)-sequence over F64, using
(61−37, 61, 513)-Net in Base 64
(24, 61, 513)-net in base 64, using
- 3 times m-reduction [i] based on (24, 64, 513)-net in base 64, using
- base change [i] based on digital (8, 48, 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, 48, 513)-net over F256, using
(61−37, 61, 125706)-Net in Base 64 — Upper bound on s
There is no (24, 61, 125707)-net in base 64, because
- 1 times m-reduction [i] would yield (24, 60, 125707)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 2 348580 845367 197368 235681 904142 759737 386191 673980 816102 628474 395856 371164 597580 968117 824321 859055 431454 134064 > 6460 [i]