MinT - Best Known (99, ∞, s)-Nets in Base 4

Best Known (99, ∞, s)-Nets in Base 4

Digital (99, m, 104)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (99, 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 (99, m, 144)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (99, 143)-sequence over F₄, using
- t-expansion [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

There is no (99, m, 314)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (99, 1251, 314)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹²⁵¹, 314, S₄, 4, 1152), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 936265 976839 135801 376683 182545 735273 078578 714663 700704 195210 822558 057623 079435 874376 095003 077698 366007 235062 325558 249581 113142 096243 534728 638626 466111 155778 266103 711165 250768 179924 193697 701511 201522 892381 013581 090099 572047 269760 268004 920035 132686 368895 909608 075268 158620 211765 080209 255421 307048 246325 284447 628357 413082 699678 086189 916744 921325 497396 136188 781663 243847 051270 631226 813494 075248 229837 173573 238235 040729 192193 992682 014580 356612 687603 576704 490261 884993 975680 745207 628152 407458 242342 209940 435906 432966 908997 660543 128705 191253 567313 650574 557126 935750 486174 983613 475885 313948 629553 280869 487662 432403 231304 993376 951286 820860 628683 769282 323016 924506 697701 794173 992285 928637 570792 364660 364554 730647 406676 595516 649230 056276 689292 494514 244855 609952 305152 / 1153 > 4¹²⁵¹ [i]