MinT - Best Known (174−35, 174, s)-Nets in Base 4

Best Known (174−35, 174, s)-Nets in Base 4

(174−35, 174, 1062)-Net over F₄ — Constructive and digital

Digital (139, 174, 1062)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (17, 34, 34)-net over F₄, using
  - trace code for nets [i] based on digital (0, 17, 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 (105, 140, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 35, 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]

(174−35, 174, 5455)-Net over F₄ — Digital

Digital (139, 174, 5455)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−35, 174, 3203818)-Net in Base 4 — Upper bound on s

There is no (139, 174, 3203819)-net in base 4, because

1 times m-reduction [i] would yield (139, 173, 3203819)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 143 344106 940633 217824 069729 533490 748850 221105 724720 146433 968220 584373 585409 778847 637082 366492 107475 930780 > 4¹⁷³ [i]