MinT - Best Known (159−59, 159, s)-Nets in Base 8

Best Known (159−59, 159, s)-Nets in Base 8

(159−59, 159, 354)-Net over F₈ — Constructive and digital

Digital (100, 159, 354)-net over F₈, using

t-expansion [i] based on digital (93, 159, 354)-net over F₈, using
- 13 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(159−59, 159, 514)-Net in Base 8 — Constructive

(100, 159, 514)-net in base 8, using

1 times m-reduction [i] based on (100, 160, 514)-net in base 8, using
- base change [i] based on digital (60, 120, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 60, 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]

(159−59, 159, 973)-Net over F₈ — Digital

Digital (100, 159, 973)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−59, 159, 138755)-Net in Base 8 — Upper bound on s

There is no (100, 159, 138756)-net in base 8, because

1 times m-reduction [i] would yield (100, 158, 138756)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 48781 182944 422753 450363 143840 484719 336994 899876 978037 645300 433836 318223 131433 434321 396383 812005 191901 641310 712872 740579 854447 338894 827058 560128 > 8¹⁵⁸ [i]