MinT - Best Known (136, 136+∞, s)-Nets in Base 3

Best Known (136, 136+∞, s)-Nets in Base 3

Digital (136, m, 78)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (136, 77)-sequence over F₃, using
- t-expansion [i] based on digital (121, 77)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

Digital (136, m, 120)-net over F₃ for arbitrarily large m, using

There is no (136, m, 286)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (136, 1709, 286)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁷⁰⁹, 286, S₃, 6, 1573), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 220 908888 734760 236262 285764 017947 461324 027760 617307 274151 354773 027912 269719 456075 119934 754658 890932 071460 715015 494556 721059 920998 281012 356767 328537 394394 064924 954647 737307 299519 005844 955191 951277 686169 388849 243225 098509 354900 819350 919896 022419 968091 495107 742467 029315 148823 918974 096083 693317 573792 198657 201722 761406 420989 284247 136167 821738 565647 006328 488390 009226 374497 144845 636666 658243 727665 288641 523859 745202 462594 693522 691882 303687 343392 803430 269222 517726 813047 294849 470080 764474 196106 123840 133140 156494 481136 803468 615522 066132 578242 013729 659410 071996 454735 149151 285495 216622 546714 963153 518830 730950 349795 577532 848592 341360 460602 835122 343412 056819 330216 396236 674479 023363 156660 094059 131380 696420 499137 828633 574817 504009 631012 934314 417898 484529 505729 763309 137815 896637 888016 297678 697469 718165 970475 843697 101810 319357 / 787 > 3¹⁷⁰⁹ [i]