MinT - Best Known (89, 89+59, s)-Nets in Base 8

Best Known (89, 89+59, s)-Nets in Base 8

(89, 89+59, 354)-Net over F₈ — Constructive and digital

Digital (89, 148, 354)-net over F₈, using

16 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+59, 384)-Net in Base 8 — Constructive

(89, 148, 384)-net in base 8, using

2 times m-reduction [i] based on (89, 150, 384)-net in base 8, using
- trace code for nets [i] based on (14, 75, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(89, 89+59, 643)-Net over F₈ — Digital

Digital (89, 148, 643)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+59, 63042)-Net in Base 8 — Upper bound on s

There is no (89, 148, 63043)-net in base 8, because

1 times m-reduction [i] would yield (89, 147, 63043)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5 680640 402783 891033 131217 345291 426919 400317 489878 833845 548268 233894 957796 662938 416946 306842 945867 644928 530952 627914 790179 716775 694560 > 8¹⁴⁷ [i]