Best Known (221, ∞, s)-Nets in Base 3
(221, ∞, 112)-Net over F3 — Constructive and digital
Digital (221, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (221, 111)-sequence over F3, using
- t-expansion [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]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(221, ∞, 163)-Net over F3 — Digital
Digital (221, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (221, 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
(221, ∞, 456)-Net in Base 3 — Upper bound on s
There is no (221, m, 457)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (221, 2735, 457)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32735, 457, S3, 6, 2514), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 438327 800406 042413 613046 361494 662345 564306 998514 966897 576905 198308 318695 675467 993532 337603 401728 925824 148629 465174 481790 087364 108406 031910 017900 472208 429753 812558 321632 226359 398132 562412 319118 864946 501174 912459 292520 911354 519308 593279 271767 699484 838336 563608 641386 834781 907535 813211 408166 367895 727271 981388 478875 339900 037284 146783 781515 110230 432706 592146 625297 237041 640417 194468 291694 093897 625393 577082 586550 538023 639947 079938 500239 729268 613061 592648 843089 562683 005086 597312 871423 789959 400667 633643 533693 804849 208761 945937 827287 646868 714485 076572 437269 048851 767909 337744 228919 050908 955343 387050 816647 955607 162710 213154 714970 194882 917099 609161 575750 488304 196295 896469 381503 142586 614163 341070 831145 585921 668390 222835 579833 495079 356749 096348 572868 086603 504993 406152 613945 607497 335990 567338 271414 104766 077048 383870 096408 342873 098791 284771 728615 086762 711338 144470 187630 518392 105429 663956 494033 184047 894688 681664 466026 847988 171107 885802 414715 039446 822820 957423 146499 335387 439932 858503 905415 710066 523038 769653 888300 940052 496075 000675 975723 294773 937128 932100 021689 013407 103037 401379 849217 822353 919103 387226 240367 856983 147654 930526 989095 355989 297954 516565 966163 943073 430268 758705 724016 015141 317765 716516 495596 139343 918495 960099 838329 433926 100106 628888 564485 947148 195864 851052 712788 100593 173687 956519 160803 127357 549933 / 503 > 32735 [i]
- extracting embedded OOA [i] would yield OOA(32735, 457, S3, 6, 2514), but