MinT - Best Known (77, 77+47, s)-Nets in Base 8

Best Known (77, 77+47, s)-Nets in Base 8

(77, 77+47, 354)-Net over F₈ — Constructive and digital

Digital (77, 124, 354)-net over F₈, using

16 times m-reduction [i] based on digital (77, 140, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 70, 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]

(77, 77+47, 432)-Net in Base 8 — Constructive

(77, 124, 432)-net in base 8, using

2 times m-reduction [i] based on (77, 126, 432)-net in base 8, using
- trace code for nets [i] based on (14, 63, 216)-net in base 64, using
  - base change [i] based on digital (5, 54, 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]

(77, 77+47, 716)-Net over F₈ — Digital

Digital (77, 124, 716)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+47, 90961)-Net in Base 8 — Upper bound on s

There is no (77, 124, 90962)-net in base 8, because

1 times m-reduction [i] would yield (77, 123, 90962)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1202 566384 721655 377350 494899 737143 197033 279726 474984 295374 263118 432359 135239 555890 464558 864077 072993 178428 506144 > 8¹²³ [i]