MinT - Best Known (137, 137+∞, s)-Nets in Base 3

Best Known (137, 137+∞, s)-Nets in Base 3

Digital (137, m, 78)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (137, 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 (137, m, 120)-net over F₃ for arbitrarily large m, using

There is no (137, m, 288)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (137, 1721, 288)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁷²¹, 288, S₃, 6, 1584), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 46 879879 748656 985058 626118 838597 460630 852588 888177 543428 756332 546515 449727 786219 638512 456574 160586 116811 368601 639197 262977 563961 105640 819397 460559 362612 486765 053209 163600 656798 466069 194926 154935 627588 460419 772656 137817 400952 587103 722354 245938 410104 825711 371298 420483 187115 076729 866804 871143 566600 146355 119751 082891 525610 597428 399754 891168 089121 922357 527403 486561 477889 518931 766107 524116 974301 289192 252524 030653 469641 921579 815593 532192 515149 562242 972449 107280 071210 109516 673637 595261 750705 056803 712392 314454 564079 622541 785824 895886 228433 724027 866755 652301 735646 435429 394787 846483 962571 001255 264493 201129 615109 038388 386508 140650 984681 085536 278623 596880 239711 041721 476105 977401 540281 965373 668342 287973 752759 146396 116413 812049 012733 351346 497139 247291 709458 635302 744937 447430 856293 162125 582817 250940 428182 018373 641923 316784 096253 / 317 > 3¹⁷²¹ [i]