MinT - Best Known (59, 59+37, s)-Nets in Base 8

Best Known (59, 59+37, s)-Nets in Base 8

(59, 59+37, 354)-Net over F₈ — Constructive and digital

Digital (59, 96, 354)-net over F₈, using

8 times m-reduction [i] based on digital (59, 104, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 52, 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]

(59, 59+37, 384)-Net in Base 8 — Constructive

(59, 96, 384)-net in base 8, using

2 times m-reduction [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]

(59, 59+37, 540)-Net over F₈ — Digital

Digital (59, 96, 540)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+37, 62988)-Net in Base 8 — Upper bound on s

There is no (59, 96, 62989)-net in base 8, because

1 times m-reduction [i] would yield (59, 95, 62989)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 62 167353 215240 163983 535923 642422 380089 818725 817806 576018 543228 086995 379276 768851 464640 > 8⁹⁵ [i]