MinT - Best Known (140−61, 140, s)-Nets in Base 8

Best Known (140−61, 140, s)-Nets in Base 8

(140−61, 140, 354)-Net over F₈ — Constructive and digital

Digital (79, 140, 354)-net over F₈, using

4 times m-reduction [i] based on digital (79, 144, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 72, 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]

(140−61, 140, 418)-Net over F₈ — Digital

Digital (79, 140, 418)-net over F₈, using

trace code for nets [i] based on digital (9, 70, 209)-net over F₆₄, using
- net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
    - a curve from the manYPoints database [i]

(140−61, 140, 26284)-Net in Base 8 — Upper bound on s

There is no (79, 140, 26285)-net in base 8, because

1 times m-reduction [i] would yield (79, 139, 26285)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338709 064420 852391 988875 605855 706425 243398 462682 701823 933563 539710 724652 378182 000820 109942 202559 360534 711477 651401 136735 082320 > 8¹³⁹ [i]