Best Known (45, ∞, s)-Nets in Base 4
(45, ∞, 56)-Net over F4 — Constructive and digital
Digital (45, m, 56)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (45, 55)-sequence over F4, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
(45, ∞, 80)-Net over F4 — Digital
Digital (45, m, 80)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (45, 79)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 45 and N(F) ≥ 80, using
(45, ∞, 149)-Net in Base 4 — Upper bound on s
There is no (45, m, 150)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (45, 595, 150)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4595, 150, S4, 4, 550), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 13 855495 187560 408700 476092 989792 355115 693220 693603 242671 394714 000697 471024 983592 178108 913338 557130 905066 376598 908531 590173 626933 679329 031918 547936 077508 264738 280694 199848 180161 961890 854230 981857 518485 215899 709548 451785 981114 287333 036085 430452 358676 236154 093194 540325 779746 887457 356392 517360 279684 103107 274056 770557 913015 237327 339329 446467 563960 149153 496302 616576 / 551 > 4595 [i]
- extracting embedded OOA [i] would yield OOA(4595, 150, S4, 4, 550), but