MinT - Best Known (165−33, 165, s)-Nets in Base 4

Best Known (165−33, 165, s)-Nets in Base 4

(165−33, 165, 1062)-Net over F₄ — Constructive and digital

Digital (132, 165, 1062)-net over F₄, using

4¹ times duplication [i] based on digital (131, 164, 1062)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (16, 32, 34)-net over F₄, using
    - trace code for nets [i] based on digital (0, 16, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
        Niederreiter sequence [i]
  2. digital (99, 132, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 33, 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]

(165−33, 165, 5437)-Net over F₄ — Digital

Digital (132, 165, 5437)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−33, 165, 3361481)-Net in Base 4 — Upper bound on s

There is no (132, 165, 3361482)-net in base 4, because

1 times m-reduction [i] would yield (132, 164, 3361482)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 546 814032 317570 102681 407176 576585 113407 424807 773948 456159 479890 425013 277897 561843 837378 577911 906556 > 4¹⁶⁴ [i]