Best Known (175, ∞, s)-Nets in Base 3
(175, ∞, 98)-Net over F3 — Constructive and digital
Digital (175, m, 98)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (175, 97)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 98 from GarcÃa–Stichtenoth tower as constant field extension [i]
(175, ∞, 163)-Net over F3 — Digital
Digital (175, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (175, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(175, ∞, 364)-Net in Base 3 — Upper bound on s
There is no (175, m, 365)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (175, 2183, 365)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32183, 365, S3, 6, 2008), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 9 825138 581349 698219 734884 060220 711279 898867 091626 398228 250898 936072 078264 126232 649163 774973 568623 491408 210444 496072 695398 364522 610628 101107 875296 679056 744514 924394 583024 669949 602834 734532 460589 853269 377416 840490 498210 105294 883639 179139 807209 063770 476857 413993 786461 723802 553348 273236 065402 584796 929785 726261 841576 113759 351035 519287 248408 766727 853632 533865 340403 172073 243773 011408 972509 741498 641235 956473 850065 788325 160708 175289 864534 507703 841980 706333 853391 419369 189529 558578 318345 909085 342887 895368 583155 567862 441851 374679 719082 691791 177496 060442 065608 694577 241223 160455 783228 098109 744739 008264 155102 685259 829672 609786 613765 223072 994874 334248 418750 538002 873477 103858 182589 164128 256935 740854 421638 854607 052799 798566 838668 233249 185561 151121 187094 822344 129444 047079 338285 678771 094093 653272 323984 989059 169966 990061 683933 220999 563335 326481 034759 870070 843522 643090 186533 816000 761215 496839 514163 353648 015320 358017 138783 172431 247060 135752 841098 845125 181612 331554 687696 843407 576361 397937 356739 519822 016458 705584 109818 722011 540607 057421 477582 461566 417391 / 2009 > 32183 [i]
- extracting embedded OOA [i] would yield OOA(32183, 365, S3, 6, 2008), but