MinT - Best Known (75, 75+29, s)-Nets in Base 4

Best Known (75, 75+29, s)-Nets in Base 4

(75, 75+29, 384)-Net over F₄ — Constructive and digital

Digital (75, 104, 384)-net over F₄, using

1 times m-reduction [i] based on digital (75, 105, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 35, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(75, 75+29, 387)-Net in Base 4 — Constructive

(75, 104, 387)-net in base 4, using

1 times m-reduction [i] based on (75, 105, 387)-net in base 4, using
- trace code for nets [i] based on (5, 35, 129)-net in base 64, using
  - base change [i] based on digital (0, 30, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(75, 75+29, 663)-Net over F₄ — Digital

Digital (75, 104, 663)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(75, 75+29, 54161)-Net in Base 4 — Upper bound on s

There is no (75, 104, 54162)-net in base 4, because

1 times m-reduction [i] would yield (75, 103, 54162)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 102 865142 648441 532269 756131 495867 251973 527115 794999 784012 570312 > 4¹⁰³ [i]