MinT - Best Known (112−73, 112, s)-Nets in Base 4

Best Known (112−73, 112, s)-Nets in Base 4

Digital (39, 112, 56)-net over F₄, using

t-expansion [i] based on digital (33, 112, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (39, 112, 66)-net over F₄, using

t-expansion [i] based on digital (37, 112, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

There is no (39, 112, 299)-net in base 4, because

1 times m-reduction [i] would yield (39, 111, 299)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹¹¹, 299, S₄, 72), but
  - 1 times code embedding in larger space [i] would yield OA(4¹¹², 300, S₄, 72), but
    - the linear programming bound shows that MÂ â‰¥ 216 705949 638772 548893 995683 517048 281707 246602 048648 981781 371507 624156 887946 346729 736732 095060 734560 449831 191166 503895 856169 237557 706276 061769 811617 718793 914039 962197 995774 171403 853869 002387 490485 484405 539612 127517 136111 726531 444736 / 6 680635 844502 955455 628139 668930 074474 571364 473300 532415 243797 229360 171593 094760 720472 142107 654322 870548 900431 119670 173133 635366 414819 375010 297936 086938 122964 329217 > 4¹¹² [i]