Best Known (222, 222+∞, s)-Nets in Base 3
(222, 222+∞, 112)-Net over F3 — Constructive and digital
Digital (222, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (222, 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
(222, 222+∞, 163)-Net over F3 — Digital
Digital (222, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (222, 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
(222, 222+∞, 458)-Net in Base 3 — Upper bound on s
There is no (222, m, 459)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (222, 2747, 459)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32747, 459, S3, 6, 2525), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 193896 719646 731863 768451 213187 677564 328450 208590 070076 960602 328735 032671 772133 870386 808902 159978 173282 793280 566419 988531 149352 122286 121622 069494 326349 898825 541095 793115 280284 669052 180708 317924 960772 652187 567155 654093 822044 893767 667337 652273 524331 773200 813438 766232 170892 021780 777894 099949 695675 177690 112513 264408 592426 020016 547353 366630 246960 808554 193418 076508 790802 850203 430123 358210 941835 415623 684807 872546 685808 128659 796068 103449 999321 171694 482564 628382 265205 675852 581480 966226 217178 110950 462059 688204 201071 520479 344813 900933 564768 185752 783556 864158 363497 571468 722056 987454 217249 198171 542623 727856 819013 404431 991260 337008 673114 353860 531841 103612 091310 412342 491342 451900 920637 628241 222777 175692 353475 160817 453477 518653 468867 164473 606345 732922 293578 162282 063059 640394 988055 114508 713791 571300 554844 753643 841068 279412 542455 806007 181737 188443 225241 004271 564718 467409 105472 681629 115781 209554 626984 728306 240837 180060 852448 104950 355001 016389 103122 120758 486187 766898 384247 743647 465831 728645 759386 003942 864374 815118 877127 578922 483228 501357 620483 847703 642195 376521 422313 688620 365007 202407 724260 467246 243597 512477 781137 278960 088361 409711 576947 123833 361030 751687 138833 570561 769670 100170 280316 712461 740703 924756 066682 232832 804974 128428 642243 337350 372501 133724 003342 932645 318915 583133 538839 864172 125937 987886 924694 031282 819984 / 421 > 32747 [i]
- extracting embedded OOA [i] would yield OOA(32747, 459, S3, 6, 2525), but