MinT - Best Known (84, ∞, s)-Nets in Base 4

Best Known (84, ∞, s)-Nets in Base 4

Digital (84, m, 104)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (84, 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 (84, m, 129)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (84, 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 (84, m, 268)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (84, 1067, 268)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁰⁶⁷, 268, S₄, 4, 983), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 105 017103 884449 630743 249610 084094 838137 512194 206937 817598 365894 277005 170912 550582 031186 543350 888200 295290 161614 648770 612011 323026 105870 825770 471041 081228 242810 341595 479154 605841 109748 260791 543364 822871 047322 395646 610791 716977 837321 918843 851224 630741 783010 981224 081288 464677 492260 357967 135597 225218 766295 142876 419168 775926 503381 654065 119818 435413 434783 403456 442440 385935 569241 255106 150337 050632 426896 998820 667223 032780 928263 953415 649575 466029 882171 095561 683208 208167 092426 348363 629236 455216 306691 024036 180118 109517 472105 514305 390523 891113 166171 840119 637412 195874 968220 766372 500505 019150 095479 808182 388351 097750 109912 560634 206913 776115 762990 153728 / 41 > 4¹⁰⁶⁷ [i]