MinT - Best Known (136−41, 136, s)-Nets in Base 8

Best Known (136−41, 136, s)-Nets in Base 8

(136−41, 136, 484)-Net over F₈ — Constructive and digital

Digital (95, 136, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (20, 40, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 20, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (55, 96, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 48, 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]

(136−41, 136, 576)-Net in Base 8 — Constructive

(95, 136, 576)-net in base 8, using

14 times m-reduction [i] based on (95, 150, 576)-net in base 8, using
- trace code for nets [i] based on (20, 75, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(136−41, 136, 2670)-Net over F₈ — Digital

Digital (95, 136, 2670)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−41, 136, 1479319)-Net in Base 8 — Upper bound on s

There is no (95, 136, 1479320)-net in base 8, because

1 times m-reduction [i] would yield (95, 135, 1479320)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 632580 437831 058060 043852 188429 336917 392359 736687 943677 765397 149367 655141 733942 702892 485942 361585 182772 764505 037483 362163 > 8¹³⁵ [i]