MinT - Best Known (89, ∞, s)-Nets in Base 4

Best Known (89, ∞, s)-Nets in Base 4

Digital (89, m, 104)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (89, 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 (89, m, 129)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (89, 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 (89, m, 284)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (89, 1130, 284)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹¹³⁰, 284, S₄, 4, 1041), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 129328 362830 984652 127866 858240 355444 150359 558347 234163 225217 181080 698990 326898 025421 343053 590562 069572 401606 333791 534688 499746 939211 729552 170214 331844 509801 486813 807277 295293 801670 835128 767941 363199 196041 280140 796511 038145 802948 578701 823223 837317 939123 873183 661661 838862 218087 014835 498081 780369 570354 145848 716892 336533 770359 265289 206008 193892 652061 783916 695134 617501 485889 548641 395616 398588 643363 142543 957806 963099 579331 363686 827875 786499 813730 265423 454726 358236 413796 421094 421383 760617 723164 221159 051017 757756 724357 015920 957236 098667 962094 532139 073906 936329 926852 800478 262720 572160 723316 227180 133212 885416 616724 488372 636747 915728 330267 094584 325829 726860 802050 524341 789528 355040 657408 / 521 > 4¹¹³⁰ [i]