MinT - Best Known (85, 85+∞, s)-Nets in Base 4

Best Known (85, 85+∞, s)-Nets in Base 4

Digital (85, m, 104)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (85, 103)-sequence over F₄, using
- t-expansion [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (85, m, 129)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (85, 128)-sequence over F₄, using
- t-expansion [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

There is no (85, m, 271)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (85, 1079, 271)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁰⁷⁹, 271, S₄, 4, 994), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 42117 671764 431092 811527 100173 757709 832708 383428 169510 530334 557112 851273 901401 079489 656836 085418 571159 956385 985243 016034 450829 178527 877167 783279 936054 202341 286411 099358 641760 020702 965856 715041 671929 050231 863770 228910 750386 881310 384251 065563 650394 391738 912751 519472 133401 017225 443388 458746 729409 529568 403444 769421 425807 108362 989242 277393 915066 923811 848612 507045 165228 039339 639820 642835 597183 427338 459661 419093 894809 545363 738607 401765 806446 035512 657656 554090 012136 002805 781517 281671 563354 760363 268901 565196 438072 728615 010222 405852 316447 834846 037764 052948 599926 561751 558006 305939 174822 538732 129488 261346 541649 995092 163306 618016 169681 898354 187556 605422 206976 / 995 > 4¹⁰⁷⁹ [i]