MinT - Best Known (166−33, 166, s)-Nets in Base 4

Best Known (166−33, 166, s)-Nets in Base 4

(166−33, 166, 1076)-Net over F₄ — Constructive and digital

Digital (133, 166, 1076)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (18, 34, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 17, 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 (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]

(166−33, 166, 5677)-Net over F₄ — Digital

Digital (133, 166, 5677)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−33, 166, 3665722)-Net in Base 4 — Upper bound on s

There is no (133, 166, 3665723)-net in base 4, because

1 times m-reduction [i] would yield (133, 165, 3665723)-net in base 4, but
- 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]