Best Known (163, ∞, s)-Nets in Base 3
(163, ∞, 86)-Net over F3 — Constructive and digital
Digital (163, m, 86)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (163, 85)-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) ≥ 86 from GarcÃa–Stichtenoth tower as constant field extension [i]
(163, ∞, 163)-Net over F3 — Digital
Digital (163, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (163, 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
(163, ∞, 340)-Net in Base 3 — Upper bound on s
There is no (163, m, 341)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (163, 1698, 341)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31698, 341, S3, 5, 1535), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 412 841647 956125 501551 539552 836634 023339 531359 300868 819955 980196 306904 137881 347289 437733 529647 653657 281173 366083 422859 103726 417467 971986 216773 456327 293792 124662 448940 286761 091501 735325 551893 979066 411928 087844 527509 885600 445972 329936 109056 150426 753710 224877 178117 721123 829302 717709 509146 561004 909573 340976 444161 920486 407240 784099 039327 352735 797308 839490 901856 829206 293398 894472 588800 709864 195113 577858 426835 060483 769600 370493 118210 004330 340660 599295 498701 522265 967646 113931 961094 808035 327754 811611 846681 451633 431032 219735 821986 687997 172504 548473 220198 411482 138064 073387 898475 940594 082328 367381 651558 331564 857989 097246 077546 732547 523651 372295 389141 431921 111370 829370 155270 966437 800149 223819 760571 912611 409929 085860 273990 799082 680976 115120 113336 639805 422117 795750 286487 970738 848935 006944 516058 531557 170846 133410 592699 / 256 > 31698 [i]
- extracting embedded OOA [i] would yield OOA(31698, 341, S3, 5, 1535), but