Best Known (230, 230+∞, s)-Nets in Base 3
(230, 230+∞, 112)-Net over F3 — Constructive and digital
Digital (230, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (230, 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
(230, 230+∞, 163)-Net over F3 — Digital
Digital (230, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (230, 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
(230, 230+∞, 475)-Net in Base 3 — Upper bound on s
There is no (230, m, 476)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (230, 2373, 476)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32373, 476, S3, 5, 2143), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4 511714 948975 674396 230017 394633 305019 416229 693788 578444 103323 393365 730624 474921 501028 464542 326766 266392 714123 551128 161791 724068 460632 201971 533148 500109 980062 417588 285266 856507 504511 766825 938118 884361 359796 293530 635020 872744 903759 026364 091152 809951 434214 969843 419712 047792 111350 201311 896966 038613 451164 024143 124740 963111 402105 369771 208520 461802 787822 767036 389161 031631 743182 712811 536011 840260 045799 375854 592533 005031 588721 783828 614069 331345 031922 488768 674016 534096 832357 744508 565305 591847 910980 249934 142167 546015 522560 316460 791924 529102 208182 295839 138219 129447 337307 346638 600392 409857 796122 402526 852022 874869 953835 794129 516745 621179 694473 500941 875607 116156 907189 551541 010602 573087 027020 686064 846973 213874 664801 308349 870686 579174 153519 974869 973153 497884 959642 459695 188937 544681 533323 379442 738948 406670 853233 802437 424747 624823 466962 962654 702836 139086 134557 973724 304717 123212 494510 614449 174557 566369 524264 056254 257723 051043 672945 845116 817399 241881 215753 013350 985098 291005 790216 987875 704855 350928 268506 922934 199044 688255 020538 253894 263360 824115 088643 412268 923283 975045 170895 418439 113787 470009 609848 412448 041062 034133 657586 193548 763612 802517 / 268 > 32373 [i]
- extracting embedded OOA [i] would yield OOA(32373, 476, S3, 5, 2143), but