MinT - Best Known (189−37, 189, s)-Nets in Base 4

Best Known (189−37, 189, s)-Nets in Base 4

(189−37, 189, 1094)-Net over F₄ — Constructive and digital

Digital (152, 189, 1094)-net over F₄, using

4¹ times duplication [i] based on digital (151, 188, 1094)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (22, 40, 66)-net over F₄, using
    - trace code for nets [i] based on digital (2, 20, 33)-net over F₁₆, using
      - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
  2. digital (111, 148, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 37, 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]

(189−37, 189, 6911)-Net over F₄ — Digital

Digital (152, 189, 6911)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−37, 189, 4888657)-Net in Base 4 — Upper bound on s

There is no (152, 189, 4888658)-net in base 4, because

1 times m-reduction [i] would yield (152, 188, 4888658)-net in base 4, but
- 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]