MinT - Best Known (74−31, 74, s)-Nets in Base 8

Best Known (74−31, 74, s)-Nets in Base 8

(74−31, 74, 256)-Net over F₈ — Constructive and digital

Digital (43, 74, 256)-net over F₈, using

2 times m-reduction [i] based on digital (43, 76, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 38, 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]

(74−31, 74, 258)-Net in Base 8 — Constructive

(43, 74, 258)-net in base 8, using

trace code for nets [i] based on (6, 37, 129)-net in base 64, using
- 5 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]

(74−31, 74, 322)-Net over F₈ — Digital

Digital (43, 74, 322)-net over F₈, using

trace code for nets [i] based on digital (6, 37, 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]

(74−31, 74, 22778)-Net in Base 8 — Upper bound on s

There is no (43, 74, 22779)-net in base 8, because

1 times m-reduction [i] would yield (43, 73, 22779)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 842567 410831 555464 372081 593372 761723 189076 578150 728045 071076 445872 > 8⁷³ [i]