MinT - Best Known (27−8, 27, s)-Nets in Base 8

Best Known (27−8, 27, s)-Nets in Base 8

(27−8, 27, 333)-Net over F₈ — Constructive and digital

Digital (19, 27, 333)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 3, 73)-net over F₈, using
  - kÂ =Â 2 construction [i]
2. digital (4, 8, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 4, 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]
3. digital (8, 16, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 8, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(27−8, 27, 528)-Net in Base 8 — Constructive

(19, 27, 528)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. (14, 22, 514)-net in base 8, using
  - trace code for nets [i] based on (3, 11, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
      - base change [i] based on digital (0, 9, 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]

(27−8, 27, 1473)-Net over F₈ — Digital

Digital (19, 27, 1473)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(27−8, 27, 394284)-Net in Base 8 — Upper bound on s

There is no (19, 27, 394285)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 417859 528984 049639 689831 > 8²⁷ [i]