MinT - Best Known (172−69, 172, s)-Nets in Base 8

Best Known (172−69, 172, s)-Nets in Base 8

(172−69, 172, 354)-Net over F₈ — Constructive and digital

Digital (103, 172, 354)-net over F₈, using

t-expansion [i] based on digital (93, 172, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 86, 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]

(172−69, 172, 384)-Net in Base 8 — Constructive

(103, 172, 384)-net in base 8, using

t-expansion [i] based on (102, 172, 384)-net in base 8, using
- trace code for nets [i] based on (16, 86, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (16, 91, 192)-net in base 64, using
    - base change [i] based on digital (3, 78, 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]

(172−69, 172, 708)-Net over F₈ — Digital

Digital (103, 172, 708)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−69, 172, 67336)-Net in Base 8 — Upper bound on s

There is no (103, 172, 67337)-net in base 8, because

1 times m-reduction [i] would yield (103, 171, 67337)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26821 238919 650794 376752 238070 144890 779652 531839 027916 101623 953114 019966 831477 117078 009868 094691 706487 907733 620398 944526 704215 412693 579077 721881 546632 830176 > 8¹⁷¹ [i]