MinT - Best Known (99−37, 99, s)-Nets in Base 8

Best Known (99−37, 99, s)-Nets in Base 8

(99−37, 99, 354)-Net over F₈ — Constructive and digital

Digital (62, 99, 354)-net over F₈, using

11 times m-reduction [i] based on digital (62, 110, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 55, 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]

(99−37, 99, 432)-Net in Base 8 — Constructive

(62, 99, 432)-net in base 8, using

8¹ times duplication [i] based on (61, 98, 432)-net in base 8, using
- trace code for nets [i] based on (12, 49, 216)-net in base 64, using
  - base change [i] based on digital (5, 42, 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]

(99−37, 99, 639)-Net over F₈ — Digital

Digital (62, 99, 639)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−37, 99, 89084)-Net in Base 8 — Upper bound on s

There is no (62, 99, 89085)-net in base 8, because

1 times m-reduction [i] would yield (62, 98, 89085)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 31833 005848 840687 168155 801989 738353 719289 032203 432220 194911 220047 722074 457159 241945 420946 > 8⁹⁸ [i]