MinT - Best Known (70, 70+43, s)-Nets in Base 8

Best Known (70, 70+43, s)-Nets in Base 8

(70, 70+43, 354)-Net over F₈ — Constructive and digital

Digital (70, 113, 354)-net over F₈, using

13 times m-reduction [i] based on digital (70, 126, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 63, 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+43, 432)-Net in Base 8 — Constructive

(70, 113, 432)-net in base 8, using

8¹ times duplication [i] based on (69, 112, 432)-net in base 8, using
- trace code for nets [i] based on (13, 56, 216)-net in base 64, using
  - base change [i] based on digital (5, 48, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(70, 70+43, 653)-Net over F₈ — Digital

Digital (70, 113, 653)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+43, 81246)-Net in Base 8 — Upper bound on s

There is no (70, 113, 81247)-net in base 8, because

1 times m-reduction [i] would yield (70, 112, 81247)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 139999 999399 178693 956524 160993 135424 206015 913551 523514 964549 497049 798140 829347 333879 246911 834103 905490 > 8¹¹² [i]