Best Known (172, ∞, s)-Nets in Base 3
(172, ∞, 95)-Net over F3 — Constructive and digital
Digital (172, m, 95)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (172, 94)-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) ≥ 95 from GarcÃa–Stichtenoth tower as constant field extension [i]
(172, ∞, 163)-Net over F3 — Digital
Digital (172, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (172, 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
(172, ∞, 358)-Net in Base 3 — Upper bound on s
There is no (172, m, 359)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (172, 1788, 359)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31788, 359, S3, 5, 1616), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 6872 209995 967174 543807 502784 399508 581930 153971 986932 219857 505786 279575 413692 766179 579356 623841 495740 111154 157440 351836 548982 761200 896683 972827 392019 237467 000842 469081 576393 309621 241937 220799 450403 795766 240036 589646 508002 053928 255649 239992 061725 721516 589744 257707 354317 639739 862370 786618 966612 168519 199284 759676 233587 462259 786376 494769 572737 571581 728567 347733 024512 148106 937278 068931 872046 753387 556062 435240 148479 765545 246193 693750 480471 385463 272481 956475 217227 344440 539018 718919 417626 401097 978078 189448 696833 514718 781916 585507 968262 230297 257746 358768 113091 381286 968805 226422 283719 504908 622963 319803 690932 767850 391782 151047 673394 606519 440520 093533 306012 980270 813022 348496 146271 277019 236409 838708 349417 764000 413831 451722 939073 343243 613239 040375 262310 364500 861552 183844 146317 343197 404222 352604 209995 166392 109294 932708 378991 434912 509823 351793 643363 545037 159355 / 539 > 31788 [i]
- extracting embedded OOA [i] would yield OOA(31788, 359, S3, 5, 1616), but