MinT - Best Known (56−25, 56, s)-Nets in Base 8

Best Known (56−25, 56, s)-Nets in Base 8

(56−25, 56, 208)-Net over F₈ — Constructive and digital

Digital (31, 56, 208)-net over F₈, using

trace code for nets [i] based on digital (3, 28, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(56−25, 56, 226)-Net over F₈ — Digital

Digital (31, 56, 226)-net over F₈, using

trace code for nets [i] based on digital (3, 28, 113)-net over F₆₄, using
- net from sequence [i] based on digital (3, 112)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 113, using
    - a curve from the manYPoints database [i]

(56−25, 56, 10402)-Net in Base 8 — Upper bound on s

There is no (31, 56, 10403)-net in base 8, because

1 times m-reduction [i] would yield (31, 55, 10403)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 46 811735 444606 754407 967914 218276 319802 578702 686808 > 8⁵⁵ [i]