MinT - Best Known (44, 44+31, s)-Nets in Base 8

Best Known (44, 44+31, s)-Nets in Base 8

(44, 44+31, 256)-Net over F₈ — Constructive and digital

Digital (44, 75, 256)-net over F₈, using

3 times m-reduction [i] based on digital (44, 78, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 39, 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]

(44, 44+31, 258)-Net in Base 8 — Constructive

(44, 75, 258)-net in base 8, using

1 times m-reduction [i] based on (44, 76, 258)-net in base 8, using
- trace code for nets [i] based on (6, 38, 129)-net in base 64, using
  - 4 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]

(44, 44+31, 322)-Net over F₈ — Digital

Digital (44, 75, 322)-net over F₈, using

1 times m-reduction [i] based on digital (44, 76, 322)-net over F₈, using
- trace code for nets [i] based on digital (6, 38, 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]

(44, 44+31, 26167)-Net in Base 8 — Upper bound on s

There is no (44, 75, 26168)-net in base 8, because

1 times m-reduction [i] would yield (44, 74, 26168)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 742242 731640 405694 254632 761116 442278 409441 813201 808616 452724 110050 > 8⁷⁴ [i]