Best Known (45, 45+∞, s)-Nets in Base 5
(45, 45+∞, 78)-Net over F5 — Constructive and digital
Digital (45, m, 78)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (45, 77)-sequence over F5, using
- t-expansion [i] based on digital (38, 77)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 38 and N(F) ≥ 78, using
- t-expansion [i] based on digital (38, 77)-sequence over F5, using
(45, 45+∞, 88)-Net over F5 — Digital
Digital (45, m, 88)-net over F5 for arbitrarily large m, using
- net from sequence [i] based on digital (45, 87)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 45 and N(F) ≥ 88, using
(45, 45+∞, 197)-Net in Base 5 — Upper bound on s
There is no (45, m, 198)-net in base 5 for arbitrarily large m, because
- m-reduction [i] would yield (45, 590, 198)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5590, 198, S5, 3, 545), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2 837921 633145 156337 051507 240877 450523 541151 434430 855851 627361 253102 207417 128243 397731 362649 446120 796917 508636 854409 614670 376674 337527 882310 381823 302125 270813 548550 630803 733796 764294 868850 824233 538040 015856 143373 776785 297570 556727 710919 997621 698859 419565 018504 557127 493058 948203 743468 495533 168942 862405 280959 593036 440852 982311 739517 698107 328728 177951 097597 132152 838642 025756 248077 339449 764622 258953 750133 514404 296875 / 91 > 5590 [i]
- extracting embedded OOA [i] would yield OOA(5590, 198, S5, 3, 545), but