MinT - Best Known (188−36, 188, s)-Nets in Base 4

Best Known (188−36, 188, s)-Nets in Base 4

(188−36, 188, 1118)-Net over F₄ — Constructive and digital

Digital (152, 188, 1118)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (26, 44, 90)-net over F₄, using
  - trace code for nets [i] based on digital (4, 22, 45)-net over F₁₆, using
    - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
        a function field by GarcÃa and Quoos [i]
2. digital (108, 144, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 36, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(188−36, 188, 7962)-Net over F₄ — Digital

Digital (152, 188, 7962)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−36, 188, 4888657)-Net in Base 4 — Upper bound on s

There is no (152, 188, 4888658)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 153914 153972 559688 846678 159327 463875 123798 789781 580612 938616 900467 557094 776132 306899 324741 479741 150062 171077 147880 > 4¹⁸⁸ [i]