Best Known (45−31, 45, s)-Nets in Base 128
(45−31, 45, 288)-Net over F128 — Constructive and digital
Digital (14, 45, 288)-net over F128, using
- t-expansion [i] based on digital (9, 45, 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
(45−31, 45, 353)-Net over F128 — Digital
Digital (14, 45, 353)-net over F128, using
- net from sequence [i] based on digital (14, 352)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 14 and N(F) ≥ 353, using
(45−31, 45, 513)-Net in Base 128
(14, 45, 513)-net in base 128, using
- 3 times m-reduction [i] based on (14, 48, 513)-net in base 128, using
- base change [i] based on digital (8, 42, 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, 42, 513)-net over F256, using
(45−31, 45, 76748)-Net in Base 128 — Upper bound on s
There is no (14, 45, 76749)-net in base 128, because
- 1 times m-reduction [i] would yield (14, 44, 76749)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 521 496296 948371 165855 006270 295066 672197 470723 648772 879268 272525 891919 867830 111063 556172 568896 > 12844 [i]