MinT - Best Known (165−32, 165, s)-Nets in Base 4

Best Known (165−32, 165, s)-Nets in Base 4

(165−32, 165, 1094)-Net over F₄ — Constructive and digital

Digital (133, 165, 1094)-net over F₄, using

4¹ times duplication [i] based on digital (132, 164, 1094)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 36, 66)-net over F₄, using
    - trace code for nets [i] based on digital (2, 18, 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 (96, 128, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 32, 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]

(165−32, 165, 6644)-Net over F₄ — Digital

Digital (133, 165, 6644)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−32, 165, 3665722)-Net in Base 4 — Upper bound on s

There is no (133, 165, 3665723)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2187 253496 951124 508876 150710 529105 621734 128007 913425 956655 447994 074928 807596 923246 834658 229023 836785 > 4¹⁶⁵ [i]