MinT - Best Known (89, 89+69, s)-Nets in Base 8

Best Known (89, 89+69, s)-Nets in Base 8

(89, 89+69, 354)-Net over F₈ — Constructive and digital

Digital (89, 158, 354)-net over F₈, using

6 times m-reduction [i] based on digital (89, 164, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 82, 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]

(89, 89+69, 450)-Net over F₈ — Digital

Digital (89, 158, 450)-net over F₈, using

trace code for nets [i] based on digital (10, 79, 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]

(89, 89+69, 28589)-Net in Base 8 — Upper bound on s

There is no (89, 158, 28590)-net in base 8, because

1 times m-reduction [i] would yield (89, 157, 28590)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6102 022797 189013 758824 323610 549401 913643 898616 922374 620240 847913 512264 166262 004303 960969 815330 321481 529014 756061 020594 642417 555912 619512 456078 > 8¹⁵⁷ [i]