MinT - Best Known (165, 165+41, s)-Nets in Base 4

Best Known (165, 165+41, s)-Nets in Base 4

(165, 165+41, 1076)-Net over F₄ — Constructive and digital

Digital (165, 206, 1076)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (22, 42, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 21, 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 (123, 164, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 41, 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, 165+41, 6646)-Net over F₄ — Digital

Digital (165, 206, 6646)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+41, 4104858)-Net in Base 4 — Upper bound on s

There is no (165, 206, 4104859)-net in base 4, because

1 times m-reduction [i] would yield (165, 205, 4104859)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2644 232044 916148 216193 971168 559431 534656 310539 907922 933503 610166 487893 412822 095139 738726 373233 671030 975057 286394 485465 475654 > 4²⁰⁵ [i]