Best Known (219, 219+∞, s)-Nets in Base 3
(219, 219+∞, 112)-Net over F3 — Constructive and digital
Digital (219, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (219, 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
(219, 219+∞, 163)-Net over F3 — Digital
Digital (219, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (219, 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
(219, 219+∞, 452)-Net in Base 3 — Upper bound on s
There is no (219, m, 453)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (219, 2711, 453)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32711, 453, S3, 6, 2492), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 864210 199831 596173 670019 960904 079357 019224 883425 085770 715471 681748 023840 459347 477446 364026 051000 193009 073378 291754 819948 487158 651363 986317 594873 298235 904458 472288 584582 846165 043335 769647 995241 847474 784117 738691 815130 021460 260446 346685 589961 943538 867679 250633 319415 173511 237614 163153 033311 617254 472204 962005 650185 598584 724090 087731 284308 568887 558139 990356 797667 881976 994277 997648 074552 529429 069551 296707 748102 431781 235021 991975 865207 562822 967457 555728 346665 582470 614260 533501 752591 485702 582589 564163 354985 254380 048863 507107 551240 661966 646675 929954 956603 975273 851174 492355 227888 583541 599171 884510 042329 122639 306405 538032 319574 666661 684228 447901 351832 554053 504549 741747 044112 039711 999590 457621 398992 646891 495634 940464 914995 400340 656935 531651 629059 242404 169250 949941 486710 040501 753444 479949 356357 507588 553717 992506 461365 506701 935786 752847 615927 185091 728597 979889 912730 859643 083391 347017 510274 210799 303683 852363 771822 621583 548978 449497 358464 584537 491717 510140 206908 691633 319246 222860 983861 720476 913784 592957 226255 990617 544415 752706 916473 725685 795594 382637 529303 913192 438275 487686 635413 162890 983535 354897 185694 248174 825008 720195 503005 312670 513829 764024 879364 337795 666962 882263 681950 920334 686370 756049 745727 847138 318842 468622 973101 468112 609542 028928 646922 111786 492684 249335 961641 799079 857160 055547 597483 / 277 > 32711 [i]
- extracting embedded OOA [i] would yield OOA(32711, 453, S3, 6, 2492), but