MinT - Best Known (123−81, 123, s)-Nets in Base 4

Best Known (123−81, 123, s)-Nets in Base 4

Digital (42, 123, 56)-net over F₄, using

t-expansion [i] based on digital (33, 123, 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 (42, 123, 75)-net over F₄, using

t-expansion [i] based on digital (40, 123, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

There is no (42, 123, 299)-net in base 4, because

1 times m-reduction [i] would yield (42, 122, 299)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²², 299, S₄, 80), but
  - 1 times code embedding in larger space [i] would yield OA(4¹²³, 300, S₄, 80), but
    - the linear programming bound shows that MÂ â‰¥ 2748 560146 682070 152655 743458 526594 890855 641027 916664 735556 252597 572860 775913 794661 217432 495896 345810 177627 964583 890556 996538 254321 455033 196304 105698 837075 882401 095169 903277 906262 698791 724301 080160 680613 193671 354502 109472 510873 999257 526018 939146 875817 163860 345622 455358 167595 009967 768703 146004 472372 428335 711109 355929 600000 / 22 446040 732177 372937 810515 314191 518726 400135 075005 918208 508138 867552 574592 112479 389366 806684 069665 833535 324010 320306 383950 606974 097814 167392 918766 593264 929047 193248 236426 342306 695249 752371 040418 887987 592169 834860 800692 738850 873881 964401 237767 625597 > 4¹²³ [i]