MinT - Best Known (69, 69+49, s)-Nets in Base 8

Best Known (69, 69+49, s)-Nets in Base 8

(69, 69+49, 354)-Net over F₈ — Constructive and digital

Digital (69, 118, 354)-net over F₈, using

6 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+49, 450)-Net over F₈ — Digital

Digital (69, 118, 450)-net over F₈, using

trace code for nets [i] based on digital (10, 59, 225)-net over F₆₄, using
- net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(69, 69+49, 35370)-Net in Base 8 — Upper bound on s

There is no (69, 118, 35371)-net in base 8, because

1 times m-reduction [i] would yield (69, 117, 35371)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4587 552341 824064 823487 006957 883695 050402 447510 940687 912958 534118 431792 686183 593741 919019 768874 622025 167546 > 8¹¹⁷ [i]