MinT - Best Known (79−33, 79, s)-Nets in Base 8

Best Known (79−33, 79, s)-Nets in Base 8

(79−33, 79, 256)-Net over F₈ — Constructive and digital

Digital (46, 79, 256)-net over F₈, using

3 times m-reduction [i] based on digital (46, 82, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 41, 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]

(79−33, 79, 258)-Net in Base 8 — Constructive

(46, 79, 258)-net in base 8, using

1 times m-reduction [i] based on (46, 80, 258)-net in base 8, using
- trace code for nets [i] based on (6, 40, 129)-net in base 64, using
  - 2 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
    - base change [i] based on digital (0, 36, 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]

(79−33, 79, 322)-Net over F₈ — Digital

Digital (46, 79, 322)-net over F₈, using

1 times m-reduction [i] based on digital (46, 80, 322)-net over F₈, using
- trace code for nets [i] based on digital (6, 40, 161)-net over F₆₄, using
  - net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(79−33, 79, 24537)-Net in Base 8 — Upper bound on s

There is no (46, 79, 24538)-net in base 8, because

1 times m-reduction [i] would yield (46, 78, 24538)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27615 052583 256191 902426 886795 864274 054890 640075 376362 598950 392047 472347 > 8⁷⁸ [i]