Best Known (170, 170+∞, s)-Nets in Base 3
(170, 170+∞, 93)-Net over F3 — Constructive and digital
Digital (170, m, 93)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (170, 92)-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) ≥ 93 from GarcÃa–Stichtenoth tower as constant field extension [i]
(170, 170+∞, 163)-Net over F3 — Digital
Digital (170, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (170, 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
(170, 170+∞, 354)-Net in Base 3 — Upper bound on s
There is no (170, m, 355)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (170, 1768, 355)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31768, 355, S3, 5, 1598), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 992237 935931 312718 101650 718536 746182 422546 841498 857040 975362 346253 468456 811049 499053 030577 684171 412567 413760 735449 727321 441833 868014 168844 084384 813421 997207 318431 142460 294931 549085 711029 195314 131109 539847 213082 310611 021923 657743 066170 305640 428119 108165 198022 413808 230426 767582 582695 553932 090519 067884 671049 210209 868596 675840 573243 624271 230652 386713 069919 920296 374220 267274 774678 314428 117617 736259 392548 869739 127710 031956 140517 970459 578938 469759 733598 446155 262635 235316 046160 211455 290466 678004 143822 656585 745130 274324 697936 957865 815322 968529 110552 732943 213023 974520 039448 993128 604951 559411 368533 560974 180136 191897 454000 721070 067600 489053 515459 790351 905288 662728 990900 712270 046582 457818 621193 261350 409770 595589 106392 679812 044819 198700 878989 993576 640164 967334 372205 941454 644825 912976 634095 852327 373941 933836 461134 787192 093267 705032 228364 527699 971921 / 533 > 31768 [i]
- extracting embedded OOA [i] would yield OOA(31768, 355, S3, 5, 1598), but