MinT - Best Known (70, 70+41, s)-Nets in Base 8

Best Known (70, 70+41, s)-Nets in Base 8

(70, 70+41, 354)-Net over F₈ — Constructive and digital

Digital (70, 111, 354)-net over F₈, using

15 times m-reduction [i] based on digital (70, 126, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 63, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(70, 70+41, 514)-Net in Base 8 — Constructive

(70, 111, 514)-net in base 8, using

1 times m-reduction [i] based on (70, 112, 514)-net in base 8, using
- base change [i] based on digital (42, 84, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 42, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(70, 70+41, 742)-Net over F₈ — Digital

Digital (70, 111, 742)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+41, 109939)-Net in Base 8 — Upper bound on s

There is no (70, 111, 109940)-net in base 8, because

1 times m-reduction [i] would yield (70, 110, 109940)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2187 394077 820295 334005 307825 169451 856836 876302 769816 750563 566102 660684 405188 756553 910626 971216 606024 > 8¹¹⁰ [i]