MinT - Best Known (187−37, 187, s)-Nets in Base 4

Best Known (187−37, 187, s)-Nets in Base 4

(187−37, 187, 1076)-Net over F₄ — Constructive and digital

Digital (150, 187, 1076)-net over F₄, using

4¹ times duplication [i] based on digital (149, 186, 1076)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 38, 48)-net over F₄, using
    - trace code for nets [i] based on digital (1, 19, 24)-net over F₁₆, using
      - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, 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]

(187−37, 187, 6400)-Net over F₄ — Digital

Digital (150, 187, 6400)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−37, 187, 4190770)-Net in Base 4 — Upper bound on s

There is no (150, 187, 4190771)-net in base 4, because

1 times m-reduction [i] would yield (150, 186, 4190771)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9619 651779 653443 475254 892330 688574 280961 829660 418648 688062 311015 996895 156226 741818 726760 774723 709372 801339 022359 > 4¹⁸⁶ [i]