MinT - Best Known (186−107, 186, s)-Nets in Base 4

Best Known (186−107, 186, s)-Nets in Base 4

(186−107, 186, 104)-Net over F₄ — Constructive and digital

Digital (79, 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]

(186−107, 186, 112)-Net over F₄ — Digital

Digital (79, 186, 112)-net over F₄, using

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

(186−107, 186, 824)-Net in Base 4 — Upper bound on s

There is no (79, 186, 825)-net in base 4, because

1 times m-reduction [i] would yield (79, 185, 825)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2477 536275 085264 516631 035643 659742 632001 033849 163115 368723 154175 159447 561134 417020 121681 163523 965246 884137 980128 > 4¹⁸⁵ [i]