MinT - Best Known (116−31, 116, s)-Nets in Base 8

Best Known (116−31, 116, s)-Nets in Base 8

(116−31, 116, 610)-Net over F₈ — Constructive and digital

Digital (85, 116, 610)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (25, 40, 256)-net over F₈, using
  - trace code for nets [i] based on digital (5, 20, 128)-net over F₆₄, using
    - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
        a function field by SÃ©mirat [i]
2. digital (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(116−31, 116, 708)-Net in Base 8 — Constructive

(85, 116, 708)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (25, 40, 514)-net in base 8, using
  - base change [i] based on digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 15, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. digital (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(116−31, 116, 5356)-Net over F₈ — Digital

Digital (85, 116, 5356)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(116−31, 116, 7697646)-Net in Base 8 — Upper bound on s

There is no (85, 116, 7697647)-net in base 8, because

1 times m-reduction [i] would yield (85, 115, 7697647)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 71 671846 844772 310630 880491 759052 655651 747386 613255 971282 334616 636573 565014 726848 154241 829646 303741 909496 > 8¹¹⁵ [i]