Best Known (198, 198+∞, s)-Nets in Base 3
(198, 198+∞, 112)-Net over F3 — Constructive and digital
Digital (198, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (198, 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
(198, 198+∞, 163)-Net over F3 — Digital
Digital (198, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (198, 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
(198, 198+∞, 410)-Net in Base 3 — Upper bound on s
There is no (198, m, 411)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (198, 2459, 411)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32459, 411, S3, 6, 2261), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 773656 762011 143897 480606 245938 799541 176913 583636 263589 413837 902661 568249 738923 284505 955109 946759 649598 409886 659494 058944 270605 087450 233040 102895 059058 045886 286940 694931 559052 361855 315654 107937 228347 029700 087876 274094 122992 241022 981488 560976 625727 062721 884872 266292 815395 150833 754143 583872 976969 860235 847355 230638 436090 646826 784476 896104 641326 082606 670270 267684 188768 667535 727318 552433 102420 047236 282129 879508 561381 490362 192783 967469 384008 774825 476845 925401 629061 008037 008248 483699 165795 196935 961002 705010 943170 784788 177854 508309 616225 403333 738238 071858 129704 139126 816215 146131 149536 194759 333696 981689 444772 069398 481142 927624 065402 521474 402157 716447 001465 375398 453870 620577 861924 119461 273970 523894 510213 991788 929330 199286 969558 852010 981805 860818 709280 498884 062979 265582 874541 167604 281411 524346 558500 028418 379370 088987 511333 093402 461389 263397 214395 347172 274429 646549 408591 273295 470091 143121 264125 466695 213397 863210 553506 422881 876661 725786 647427 032030 432876 969036 132034 278139 842400 697475 819301 001066 032028 750820 343062 493524 922901 005497 327635 483004 524866 133086 134506 258809 077961 285238 828301 641195 977078 836019 626306 754818 730258 125948 229328 916698 422209 130593 807688 736834 926307 057748 / 377 > 32459 [i]
- extracting embedded OOA [i] would yield OOA(32459, 411, S3, 6, 2261), but