MinT - Best Known (70, 70+35, s)-Nets in Base 8

Best Known (70, 70+35, s)-Nets in Base 8

(70, 70+35, 379)-Net over F₈ — Constructive and digital

Digital (70, 105, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (49, 84, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 42, 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]

(70, 70+35, 576)-Net in Base 8 — Constructive

(70, 105, 576)-net in base 8, using

1 times m-reduction [i] based on (70, 106, 576)-net in base 8, using
- trace code for nets [i] based on (17, 53, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 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]

(70, 70+35, 1206)-Net over F₈ — Digital

Digital (70, 105, 1206)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+35, 343257)-Net in Base 8 — Upper bound on s

There is no (70, 105, 343258)-net in base 8, because

1 times m-reduction [i] would yield (70, 104, 343258)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8343 990772 633587 683039 925872 480895 318041 568471 405051 186232 425779 747575 207976 199054 815591 325917 > 8¹⁰⁴ [i]