MinT - Best Known (106−39, 106, s)-Nets in Base 8

Best Known (106−39, 106, s)-Nets in Base 8

(106−39, 106, 354)-Net over F₈ — Constructive and digital

Digital (67, 106, 354)-net over F₈, using

14 times m-reduction [i] based on digital (67, 120, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 60, 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]

(106−39, 106, 514)-Net in Base 8 — Constructive

(67, 106, 514)-net in base 8, using

8² times duplication [i] based on (65, 104, 514)-net in base 8, using
- base change [i] based on digital (39, 78, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 39, 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]

(106−39, 106, 728)-Net over F₈ — Digital

Digital (67, 106, 728)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−39, 106, 110874)-Net in Base 8 — Upper bound on s

There is no (67, 106, 110875)-net in base 8, because

1 times m-reduction [i] would yield (67, 105, 110875)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 66754 245968 895890 629021 578107 962578 679630 625265 369894 600013 567026 270098 038142 744287 888425 419776 > 8¹⁰⁵ [i]