MinT - Best Known (67, 67+45, s)-Nets in Base 8

Best Known (67, 67+45, s)-Nets in Base 8

(67, 67+45, 354)-Net over F₈ — Constructive and digital

Digital (67, 112, 354)-net over F₈, using

8 times m-reduction [i] based on digital (67, 120, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 60, 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]

(67, 67+45, 384)-Net in Base 8 — Constructive

(67, 112, 384)-net in base 8, using

trace code for nets [i] based on (11, 56, 192)-net in base 64, using
- base change [i] based on digital (3, 48, 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]

(67, 67+45, 495)-Net over F₈ — Digital

Digital (67, 112, 495)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+45, 46572)-Net in Base 8 — Upper bound on s

There is no (67, 112, 46573)-net in base 8, because

1 times m-reduction [i] would yield (67, 111, 46573)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17504 155782 906569 670312 668130 884739 989309 210636 331560 456942 325586 918827 820917 903255 796267 986739 298552 > 8¹¹¹ [i]