MinT - Best Known (69, 112, s)-Nets in Base 8

Best Known (69, 112, s)-Nets in Base 8

(69, 112, 354)-Net over F₈ — Constructive and digital

Digital (69, 112, 354)-net over F₈, using

12 times m-reduction [i] based on digital (69, 124, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 62, 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]

(69, 112, 432)-Net in Base 8 — Constructive

(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]

(69, 112, 620)-Net over F₈ — Digital

Digital (69, 112, 620)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 112, 73585)-Net in Base 8 — Upper bound on s

There is no (69, 112, 73586)-net in base 8, because

1 times m-reduction [i] would yield (69, 111, 73586)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17499 475207 296728 003279 698970 220305 338874 620552 923115 995566 158505 187799 945483 233962 993778 535715 008072 > 8¹¹¹ [i]