MinT - Best Known (190−113, 190, s)-Nets in Base 4

Best Known (190−113, 190, s)-Nets in Base 4

(190−113, 190, 104)-Net over F₄ — Constructive and digital

Digital (77, 190, 104)-net over F₄, using

t-expansion [i] based on digital (73, 190, 104)-net over F₄, using
- net from sequence [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]

(190−113, 190, 112)-Net over F₄ — Digital

Digital (77, 190, 112)-net over F₄, using

t-expansion [i] based on digital (73, 190, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(190−113, 190, 733)-Net in Base 4 — Upper bound on s

There is no (77, 190, 734)-net in base 4, because

1 times m-reduction [i] would yield (77, 189, 734)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 627743 620528 444859 700350 807482 145438 776629 783460 748354 230518 114794 868399 908951 846202 228689 752725 187017 650513 102986 > 4¹⁸⁹ [i]