MinT - Best Known (77, 77+59, s)-Nets in Base 8

Best Known (77, 77+59, s)-Nets in Base 8

(77, 77+59, 354)-Net over F₈ — Constructive and digital

Digital (77, 136, 354)-net over F₈, using

4 times m-reduction [i] based on digital (77, 140, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 70, 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]

(77, 77+59, 418)-Net over F₈ — Digital

Digital (77, 136, 418)-net over F₈, using

trace code for nets [i] based on digital (9, 68, 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]

(77, 77+59, 26654)-Net in Base 8 — Upper bound on s

There is no (77, 136, 26655)-net in base 8, because

1 times m-reduction [i] would yield (77, 135, 26655)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 690208 359313 035498 399185 532034 536134 932847 865936 333006 830407 174176 949412 321226 062028 827444 256384 315409 919302 662174 035530 > 8¹³⁵ [i]