Best Known (187, ∞, s)-Nets in Base 3
(187, ∞, 110)-Net over F3 — Constructive and digital
Digital (187, m, 110)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (187, 109)-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) ≥ 110 from GarcÃa–Stichtenoth tower as constant field extension [i]
(187, ∞, 163)-Net over F3 — Digital
Digital (187, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (187, 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
(187, ∞, 388)-Net in Base 3 — Upper bound on s
There is no (187, m, 389)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (187, 2327, 389)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32327, 389, S3, 6, 2140), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4920 856257 594501 252536 690372 841538 606421 348314 062219 820853 295585 804459 714697 099363 896194 648128 182964 224668 299116 014061 879325 163678 528382 348993 361807 209984 350888 838689 723644 901776 863279 345611 435459 143532 328941 536412 208420 719449 827859 082399 379194 838442 435164 559573 464572 518990 054971 624754 155840 543891 705326 697734 182493 063883 979118 796923 004159 934945 919035 825651 092654 710338 738145 904175 876159 818815 299209 887231 340997 531089 927508 458741 508495 670684 554681 399029 448934 304193 848896 692477 872277 100478 067513 916712 941110 942639 928584 034416 816730 032036 372624 292772 149649 539560 825390 257520 347934 834116 148595 565626 760050 031504 664086 836513 796432 262303 573846 740537 021798 794239 285224 164277 854065 770164 979438 680477 401874 624589 681082 816203 696410 257857 641045 361611 219667 339236 702198 903099 903554 061057 832252 655809 052932 889303 115896 028571 568074 469189 891110 324484 167639 367004 348833 628711 638818 033236 839473 191945 753183 203562 899424 036854 696577 161663 283462 299006 437307 704186 252673 182233 291534 573282 075530 432468 086113 543642 511676 962947 565120 791251 606199 921117 732027 089514 278187 310239 345800 866912 058678 590793 929137 971334 827872 189898 034051 643939 / 2141 > 32327 [i]
- extracting embedded OOA [i] would yield OOA(32327, 389, S3, 6, 2140), but