MinT - Best Known (133−51, 133, s)-Nets in Base 8

Best Known (133−51, 133, s)-Nets in Base 8

(133−51, 133, 354)-Net over F₈ — Constructive and digital

Digital (82, 133, 354)-net over F₈, using

17 times m-reduction [i] based on digital (82, 150, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 75, 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]

(133−51, 133, 432)-Net in Base 8 — Constructive

(82, 133, 432)-net in base 8, using

1 times m-reduction [i] based on (82, 134, 432)-net in base 8, using
- trace code for nets [i] based on (15, 67, 216)-net in base 64, using
  - 3 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 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]

(133−51, 133, 713)-Net over F₈ — Digital

Digital (82, 133, 713)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(133−51, 133, 85263)-Net in Base 8 — Upper bound on s

There is no (82, 133, 85264)-net in base 8, because

1 times m-reduction [i] would yield (82, 132, 85264)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 161410 000275 058007 063391 410499 072283 280859 162660 719436 293064 042549 209228 575936 329872 427149 163888 123949 656098 277448 874288 > 8¹³² [i]