MinT - Best Known (138−55, 138, s)-Nets in Base 8

Best Known (138−55, 138, s)-Nets in Base 8

(138−55, 138, 354)-Net over F₈ — Constructive and digital

Digital (83, 138, 354)-net over F₈, using

14 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]

(138−55, 138, 384)-Net in Base 8 — Constructive

(83, 138, 384)-net in base 8, using

2 times m-reduction [i] based on (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]

(138−55, 138, 608)-Net over F₈ — Digital

Digital (83, 138, 608)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−55, 138, 59639)-Net in Base 8 — Upper bound on s

There is no (83, 138, 59640)-net in base 8, because

1 times m-reduction [i] would yield (83, 137, 59640)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5290 466774 567027 945003 900481 019738 993032 420413 336766 522103 782834 330939 402191 104715 012569 959591 155777 944944 017428 666784 830508 > 8¹³⁷ [i]