MinT - Best Known (140−57, 140, s)-Nets in Base 8

Best Known (140−57, 140, s)-Nets in Base 8

(140−57, 140, 354)-Net over F₈ — Constructive and digital

Digital (83, 140, 354)-net over F₈, using

12 times m-reduction [i] based on digital (83, 152, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 76, 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]

(140−57, 140, 384)-Net in Base 8 — Constructive

(83, 140, 384)-net in base 8, using

trace code for nets [i] based on (13, 70, 192)-net in base 64, using
- base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(140−57, 140, 556)-Net over F₈ — Digital

Digital (83, 140, 556)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−57, 140, 49085)-Net in Base 8 — Upper bound on s

There is no (83, 140, 49086)-net in base 8, because

1 times m-reduction [i] would yield (83, 139, 49086)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338651 022225 516899 762880 610476 045505 424911 834651 513896 839643 040184 274742 650675 014657 369933 681669 704128 628913 895285 954747 247903 > 8¹³⁹ [i]