MinT - Best Known (99−39, 99, s)-Nets in Base 8

Best Known (99−39, 99, s)-Nets in Base 8

(99−39, 99, 354)-Net over F₈ — Constructive and digital

Digital (60, 99, 354)-net over F₈, using

7 times m-reduction [i] based on digital (60, 106, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 53, 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]

(99−39, 99, 384)-Net in Base 8 — Constructive

(60, 99, 384)-net in base 8, using

8¹ times duplication [i] based on (59, 98, 384)-net in base 8, using
- trace code for nets [i] based on (10, 49, 192)-net in base 64, using
  - base change [i] based on digital (3, 42, 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]

(99−39, 99, 496)-Net over F₈ — Digital

Digital (60, 99, 496)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−39, 99, 51530)-Net in Base 8 — Upper bound on s

There is no (60, 99, 51531)-net in base 8, because

1 times m-reduction [i] would yield (60, 98, 51531)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 31838 047307 554983 285778 774656 831832 421906 197796 814827 082042 839924 942238 640986 012476 371136 > 8⁹⁸ [i]