MinT - Best Known (135−50, 135, s)-Nets in Base 8

Best Known (135−50, 135, s)-Nets in Base 8

(135−50, 135, 354)-Net over F₈ — Constructive and digital

Digital (85, 135, 354)-net over F₈, using

21 times m-reduction [i] based on digital (85, 156, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 78, 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]

(135−50, 135, 514)-Net in Base 8 — Constructive

(85, 135, 514)-net in base 8, using

1 times m-reduction [i] based on (85, 136, 514)-net in base 8, using
- base change [i] based on digital (51, 102, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 51, 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]

(135−50, 135, 862)-Net over F₈ — Digital

Digital (85, 135, 862)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−50, 135, 109433)-Net in Base 8 — Upper bound on s

There is no (85, 135, 109434)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 633640 630767 505317 934032 656203 923721 423824 380473 680167 677384 537586 068017 313401 812079 507751 423477 813834 744889 073663 082737 > 8¹³⁵ [i]