MinT - Best Known (135−53, 135, s)-Nets in Base 8

Best Known (135−53, 135, s)-Nets in Base 8

(135−53, 135, 354)-Net over F₈ — Constructive and digital

Digital (82, 135, 354)-net over F₈, using

15 times m-reduction [i] based on digital (82, 150, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 75, 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]

(135−53, 135, 384)-Net in Base 8 — Constructive

(82, 135, 384)-net in base 8, using

3 times m-reduction [i] based on (82, 138, 384)-net in base 8, using
- trace code for nets [i] based on (13, 69, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
    - base change [i] based on digital (3, 60, 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]

(135−53, 135, 642)-Net over F₈ — Digital

Digital (82, 135, 642)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−53, 135, 67995)-Net in Base 8 — Upper bound on s

There is no (82, 135, 67996)-net in base 8, because

1 times m-reduction [i] would yield (82, 134, 67996)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 332123 575889 747721 192037 507233 225121 785094 306908 923673 587929 705448 697370 383924 879300 487733 190131 806717 721574 220924 030243 > 8¹³⁴ [i]