MinT - Best Known (69, 69+51, s)-Nets in Base 8

Best Known (69, 69+51, s)-Nets in Base 8

(69, 69+51, 354)-Net over F₈ — Constructive and digital

Digital (69, 120, 354)-net over F₈, using

4 times m-reduction [i] based on digital (69, 124, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 62, 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]

(69, 69+51, 418)-Net over F₈ — Digital

Digital (69, 120, 418)-net over F₈, using

trace code for nets [i] based on digital (9, 60, 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]

(69, 69+51, 28906)-Net in Base 8 — Upper bound on s

There is no (69, 120, 28907)-net in base 8, because

1 times m-reduction [i] would yield (69, 119, 28907)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 293580 297401 936277 307038 124171 365589 626256 714329 604428 793768 806919 123491 208258 670823 057193 424195 771845 887780 > 8¹¹⁹ [i]