MinT - Best Known (40−17, 40, s)-Nets in Base 8

Best Known (40−17, 40, s)-Nets in Base 8

(40−17, 40, 208)-Net over F₈ — Constructive and digital

Digital (23, 40, 208)-net over F₈, using

trace code for nets [i] based on digital (3, 20, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(40−17, 40, 226)-Net over F₈ — Digital

Digital (23, 40, 226)-net over F₈, using

trace code for nets [i] based on digital (3, 20, 113)-net over F₆₄, using
- net from sequence [i] based on digital (3, 112)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 113, using
    - a curve from the manYPoints database [i]

(40−17, 40, 258)-Net in Base 8 — Constructive

(23, 40, 258)-net in base 8, using

trace code for nets [i] based on (3, 20, 129)-net in base 64, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
  - base change [i] based on digital (0, 18, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(40−17, 40, 13583)-Net in Base 8 — Upper bound on s

There is no (23, 40, 13584)-net in base 8, because

1 times m-reduction [i] would yield (23, 39, 13584)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 166215 812560 470815 940799 866719 104195 > 8³⁹ [i]