MinT - Best Known (105, 166, s)-Nets in Base 8

Best Known (105, 166, s)-Nets in Base 8

(105, 166, 363)-Net over F₈ — Constructive and digital

Digital (105, 166, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 30, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (75, 136, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 68, 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]

(105, 166, 576)-Net in Base 8 — Constructive

(105, 166, 576)-net in base 8, using

2 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
- trace code for nets [i] based on (21, 84, 288)-net in base 64, using
  - base change [i] based on digital (9, 72, 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]

(105, 166, 1061)-Net over F₈ — Digital

Digital (105, 166, 1061)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(105, 166, 159453)-Net in Base 8 — Upper bound on s

There is no (105, 166, 159454)-net in base 8, because

1 times m-reduction [i] would yield (105, 165, 159454)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102300 672812 915546 711327 111525 759378 282370 211954 997325 503353 170644 387498 785430 957984 687739 714816 628648 301317 857257 583017 786501 312481 833562 129168 344112 > 8¹⁶⁵ [i]