MinT - Best Known (39, 39+29, s)-Nets in Base 8

Best Known (39, 39+29, s)-Nets in Base 8

(39, 39+29, 256)-Net over F₈ — Constructive and digital

Digital (39, 68, 256)-net over F₈, using

trace code for nets [i] based on digital (5, 34, 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]

(39, 39+29, 258)-Net in Base 8 — Constructive

(39, 68, 258)-net in base 8, using

trace code for nets [i] based on (5, 34, 129)-net in base 64, using
- 1 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
  - base change [i] based on digital (0, 30, 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]

(39, 39+29, 266)-Net over F₈ — Digital

Digital (39, 68, 266)-net over F₈, using

trace code for nets [i] based on digital (5, 34, 133)-net over F₆₄, using
- net from sequence [i] based on digital (5, 132)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 133, using
    - a curve from the manYPoints database [i]

(39, 39+29, 18117)-Net in Base 8 — Upper bound on s

There is no (39, 68, 18118)-net in base 8, because

1 times m-reduction [i] would yield (39, 67, 18118)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 216319 309599 014991 979352 788782 190581 254153 664676 791095 631328 > 8⁶⁷ [i]