MinT - Best Known (56, 56+35, s)-Nets in Base 8

Best Known (56, 56+35, s)-Nets in Base 8

(56, 56+35, 354)-Net over F₈ — Constructive and digital

Digital (56, 91, 354)-net over F₈, using

7 times m-reduction [i] based on digital (56, 98, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 49, 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]

(56, 56+35, 384)-Net in Base 8 — Constructive

(56, 91, 384)-net in base 8, using

1 times m-reduction [i] based on (56, 92, 384)-net in base 8, using
- trace code for nets [i] based on (10, 46, 192)-net in base 64, using
  - 3 times m-reduction [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]

(56, 56+35, 522)-Net over F₈ — Digital

Digital (56, 91, 522)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+35, 61921)-Net in Base 8 — Upper bound on s

There is no (56, 91, 61922)-net in base 8, because

1 times m-reduction [i] would yield (56, 90, 61922)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1897 605702 184328 457534 718763 794029 469339 196816 323489 133450 192232 535840 850586 231126 > 8⁹⁰ [i]