MinT - Best Known (128−43, 128, s)-Nets in Base 8

Best Known (128−43, 128, s)-Nets in Base 8

(128−43, 128, 388)-Net over F₈ — Constructive and digital

Digital (85, 128, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 28, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (57, 100, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 50, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(128−43, 128, 576)-Net in Base 8 — Constructive

(85, 128, 576)-net in base 8, using

4 times m-reduction [i] based on (85, 132, 576)-net in base 8, using
- trace code for nets [i] based on (19, 66, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(128−43, 128, 1354)-Net over F₈ — Digital

Digital (85, 128, 1354)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−43, 128, 358858)-Net in Base 8 — Upper bound on s

There is no (85, 128, 358859)-net in base 8, because

1 times m-reduction [i] would yield (85, 127, 358859)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 925450 972002 462279 163923 550710 200178 882559 429018 868747 721973 727014 123838 273780 684559 713606 199790 179577 337149 676368 > 8¹²⁷ [i]