Best Known (213, ∞, s)-Nets in Base 3
(213, ∞, 112)-Net over F3 — Constructive and digital
Digital (213, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (213, 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
(213, ∞, 163)-Net over F3 — Digital
Digital (213, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (213, 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
(213, ∞, 440)-Net in Base 3 — Upper bound on s
There is no (213, m, 441)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (213, 2639, 441)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32639, 441, S3, 6, 2426), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 115 879189 489719 132049 945883 103199 615133 978490 937512 974691 436972 955894 697513 186613 494842 559322 569878 328654 730149 348781 543492 644710 173771 286864 276488 789471 921234 039885 863870 758438 700927 723113 635785 362679 010365 649161 207942 692138 541816 464723 336862 450772 493652 987925 126106 234026 667316 705508 950042 151909 392379 431656 648216 867000 130702 072365 204388 701956 904781 322121 641193 445948 574798 008338 758070 131072 656864 964672 900250 348841 667592 986972 968279 973049 232836 745667 379901 457735 937046 147154 884703 650792 335221 099460 471454 142017 383449 897771 824154 829353 876580 041524 949621 523267 609980 680995 940924 431673 356001 507952 725073 577110 948942 843759 339221 222785 453341 223174 198978 918873 655425 081220 341216 432378 091918 044783 771310 517436 979969 222292 161326 934505 698260 253433 325499 854436 100357 216769 393998 758428 153316 058777 428980 757178 565349 724490 068705 232782 240130 825884 265487 323777 065514 068037 571934 677729 186332 452586 755532 863686 315828 861273 439675 719978 313165 380329 588296 245302 348377 164029 715871 732882 907921 769209 950912 644926 208858 463361 715456 717054 374740 455594 009621 240908 462525 864215 437483 922690 631679 291927 891261 816600 027140 081323 767267 828349 362594 299728 633161 587916 141123 999371 572694 652391 407811 241449 218776 993734 384295 328650 405183 786710 543698 120597 086576 503233 796807 271970 583788 993995 333091 / 809 > 32639 [i]
- extracting embedded OOA [i] would yield OOA(32639, 441, S3, 6, 2426), but