MinT - Best Known (75, 75+47, s)-Nets in Base 16

Best Known (75, 75+47, s)-Nets in Base 16

(75, 75+47, 563)-Net over F₁₆ — Constructive and digital

Digital (75, 122, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 28, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (47, 94, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 47, 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]

(75, 75+47, 1897)-Net over F₁₆ — Digital

Digital (75, 122, 1897)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(75, 75+47, 1358561)-Net in Base 16 — Upper bound on s

There is no (75, 122, 1358562)-net in base 16, because

1 times m-reduction [i] would yield (75, 121, 1358562)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 49 948715 738578 828033 048526 925211 264354 224518 574006 940909 104774 940584 092232 652494 836223 709110 758531 012021 733653 474542 139891 109684 911517 444021 614416 > 16¹²¹ [i]