MinT - Best Known (85, 85+57, s)-Nets in Base 8

Best Known (85, 85+57, s)-Nets in Base 8

(85, 85+57, 354)-Net over F₈ — Constructive and digital

Digital (85, 142, 354)-net over F₈, using

14 times m-reduction [i] based on digital (85, 156, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 78, 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]

(85, 85+57, 384)-Net in Base 8 — Constructive

(85, 142, 384)-net in base 8, using

8² times duplication [i] based on (83, 140, 384)-net in base 8, using
- trace code for nets [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]

(85, 85+57, 601)-Net over F₈ — Digital

Digital (85, 142, 601)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+57, 56947)-Net in Base 8 — Upper bound on s

There is no (85, 142, 56948)-net in base 8, because

1 times m-reduction [i] would yield (85, 141, 56948)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 663944 108124 414109 446636 063829 965238 869376 296855 305186 827472 855010 850425 576363 403869 244783 556792 164401 723816 318459 467798 660232 > 8¹⁴¹ [i]