MinT - Best Known (68−27, 68, s)-Nets in Base 8

Best Known (68−27, 68, s)-Nets in Base 8

(68−27, 68, 354)-Net over F₈ — Constructive and digital

Digital (41, 68, 354)-net over F₈, using

trace code for nets [i] based on digital (7, 34, 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]

(68−27, 68, 360)-Net over F₈ — Digital

Digital (41, 68, 360)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−27, 68, 36525)-Net in Base 8 — Upper bound on s

There is no (41, 68, 36526)-net in base 8, because

1 times m-reduction [i] would yield (41, 67, 36526)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 213926 653958 844996 618069 509951 018142 057452 639149 182498 357963 > 8⁶⁷ [i]