MinT - Best Known (38, 109, s)-Nets in Base 4

Best Known (38, 109, s)-Nets in Base 4

Digital (38, 109, 56)-net over F₄, using

t-expansion [i] based on digital (33, 109, 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 (38, 109, 66)-net over F₄, using

t-expansion [i] based on digital (37, 109, 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 (38, 109, 299)-net in base 4, because

2 times m-reduction [i] would yield (38, 107, 299)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁰⁷, 299, S₄, 69), but
  - 1 times code embedding in larger space [i] would yield OA(4¹⁰⁸, 300, S₄, 69), but
    - the linear programming bound shows that MÂ â‰¥ 32035 627197 915938 331057 398043 935658 055268 280947 692843 756420 308342 009262 790771 757686 789566 681959 598994 820544 645491 336699 683801 715837 288039 257637 775054 042441 190606 519006 084706 555267 060143 349882 819224 402900 263077 150720 / 298322 342823 724130 925190 654425 910734 673372 973553 804140 687925 564248 581721 629257 332957 941402 512371 132732 977046 599463 755564 990504 848088 574172 735355 806029 > 4¹⁰⁸ [i]