Best Known (189, ∞, s)-Nets in Base 3
(189, ∞, 112)-Net over F3 — Constructive and digital
Digital (189, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (189, 111)-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) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
(189, ∞, 163)-Net over F3 — Digital
Digital (189, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (189, 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
(189, ∞, 392)-Net in Base 3 — Upper bound on s
There is no (189, m, 393)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (189, 2351, 393)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32351, 393, S3, 6, 2162), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 462 234427 613671 806582 519582 699430 250303 514994 730333 063293 727043 303460 555391 142651 525539 775306 460608 508077 161631 065454 447959 845898 450569 686084 151511 117594 065464 941838 795343 152247 840714 697921 587271 145080 778002 988841 517413 121239 617547 996873 092494 710558 964525 489109 157220 216752 100519 432171 288304 991492 585205 909548 433956 379514 824442 136847 209935 202449 397709 788817 717794 573755 729242 247829 558022 791753 279519 045086 882864 945804 905662 146413 990785 485062 864477 854721 389810 755877 056518 888763 875484 432271 185061 365043 664723 395145 626930 515086 892715 866660 477592 038565 021580 150791 994368 404433 831059 406732 436289 739258 298397 438133 107103 209586 511480 028147 652790 525651 298240 440980 315861 668052 595441 206629 439425 036553 298851 722324 380994 880514 995154 477378 346860 188145 145037 587020 911323 167287 603721 966597 839301 401576 682295 341464 209402 204324 098503 268624 723389 873283 807112 982255 148987 198335 944391 514479 556843 968403 374771 517858 899623 209884 395776 212752 033051 862995 268952 146133 024680 898756 833152 371997 218657 351253 144759 875554 820456 340479 767187 923490 716532 333981 827415 052744 333561 822160 866050 425525 244171 315659 082331 910539 703713 167474 094814 449195 417396 094059 / 721 > 32351 [i]
- extracting embedded OOA [i] would yield OOA(32351, 393, S3, 6, 2162), but