MinT - Best Known (69−29, 69, s)-Nets in Base 9

Best Known (69−29, 69, s)-Nets in Base 9

(69−29, 69, 320)-Net over F₉ — Constructive and digital

Digital (40, 69, 320)-net over F₉, using

1 times m-reduction [i] based on digital (40, 70, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 35, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(69−29, 69, 334)-Net over F₉ — Digital

Digital (40, 69, 334)-net over F₉, using

1 times m-reduction [i] based on digital (40, 70, 334)-net over F₉, using
- trace code for nets [i] based on digital (5, 35, 167)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
      - a curve from the manYPoints database [i]

(69−29, 69, 32594)-Net in Base 9 — Upper bound on s

There is no (40, 69, 32595)-net in base 9, because

1 times m-reduction [i] would yield (40, 68, 32595)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 77357 756159 552544 112832 989307 270348 748762 038910 902508 294054 985425 > 9⁶⁸ [i]