MinT - Best Known (56−29, 56, s)-Nets in Base 8

Best Known (56−29, 56, s)-Nets in Base 8

(56−29, 56, 70)-Net over F₈ — Constructive and digital

Digital (27, 56, 70)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (9, 38, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(56−29, 56, 97)-Net over F₈ — Digital

Digital (27, 56, 97)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−29, 56, 3040)-Net in Base 8 — Upper bound on s

There is no (27, 56, 3041)-net in base 8, because

1 times m-reduction [i] would yield (27, 55, 3041)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 46 789685 499321 436814 596410 581081 070496 601284 349232 > 8⁵⁵ [i]