MinT - Best Known (77, 77+35, s)-Nets in Base 8

Best Known (77, 77+35, s)-Nets in Base 8

(77, 77+35, 402)-Net over F₈ — Constructive and digital

Digital (77, 112, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 28, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (49, 84, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 42, 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]

(77, 77+35, 576)-Net in Base 8 — Constructive

(77, 112, 576)-net in base 8, using

6 times m-reduction [i] based on (77, 118, 576)-net in base 8, using
- trace code for nets [i] based on (18, 59, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(77, 77+35, 1842)-Net over F₈ — Digital

Digital (77, 112, 1842)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+35, 808142)-Net in Base 8 — Upper bound on s

There is no (77, 112, 808143)-net in base 8, because

1 times m-reduction [i] would yield (77, 111, 808143)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17498 188897 296766 212167 430561 792778 476752 933566 880398 826728 745434 861658 499070 591465 877013 277642 708600 > 8¹¹¹ [i]