MinT - Best Known (186−105, 186, s)-Nets in Base 4

Best Known (186−105, 186, s)-Nets in Base 4

(186−105, 186, 104)-Net over F₄ — Constructive and digital

Digital (81, 186, 104)-net over F₄, using

t-expansion [i] based on digital (73, 186, 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]

(186−105, 186, 129)-Net over F₄ — Digital

Digital (81, 186, 129)-net over F₄, using

net from sequence [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]

(186−105, 186, 892)-Net in Base 4 — Upper bound on s

There is no (81, 186, 893)-net in base 4, because

1 times m-reduction [i] would yield (81, 185, 893)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2445 063546 068728 316258 514778 917565 952137 465647 672937 704955 233859 382125 014958 922046 747314 795802 856497 872695 071463 > 4¹⁸⁵ [i]