MinT - Best Known (67, 67+41, s)-Nets in Base 8

Best Known (67, 67+41, s)-Nets in Base 8

(67, 67+41, 354)-Net over F₈ — Constructive and digital

Digital (67, 108, 354)-net over F₈, using

12 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+41, 432)-Net in Base 8 — Constructive

(67, 108, 432)-net in base 8, using

trace code for nets [i] based on (13, 54, 216)-net in base 64, using
- 2 times m-reduction [i] based on (13, 56, 216)-net in base 64, using
  - base change [i] based on digital (5, 48, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(67, 67+41, 638)-Net over F₈ — Digital

Digital (67, 108, 638)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+41, 80477)-Net in Base 8 — Upper bound on s

There is no (67, 108, 80478)-net in base 8, because

1 times m-reduction [i] would yield (67, 107, 80478)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 272806 321757 855782 321432 650247 378089 873188 480377 567595 117992 615474 356181 926683 115859 651664 121721 > 8¹⁰⁷ [i]