MinT - Best Known (85−31, 85, s)-Nets in Base 8

Best Known (85−31, 85, s)-Nets in Base 8

(85−31, 85, 354)-Net over F₈ — Constructive and digital

Digital (54, 85, 354)-net over F₈, using

9 times m-reduction [i] based on digital (54, 94, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 47, 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]

(85−31, 85, 514)-Net in Base 8 — Constructive

(54, 85, 514)-net in base 8, using

1 times m-reduction [i] based on (54, 86, 514)-net in base 8, using
- trace code for nets [i] based on (11, 43, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (11, 44, 257)-net in base 64, using
    - base change [i] based on digital (0, 33, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(85−31, 85, 638)-Net over F₈ — Digital

Digital (54, 85, 638)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−31, 85, 104697)-Net in Base 8 — Upper bound on s

There is no (54, 85, 104698)-net in base 8, because

1 times m-reduction [i] would yield (54, 84, 104698)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 7237 950535 779983 650586 116374 830591 273044 902281 604752 247597 444986 662946 820856 > 8⁸⁴ [i]