MinT - Best Known (55−28, 55, s)-Nets in Base 8

Best Known (55−28, 55, s)-Nets in Base 8

(55−28, 55, 70)-Net over F₈ — Constructive and digital

Digital (27, 55, 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, 37, 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]

(55−28, 55, 76)-Net in Base 8 — Constructive

(27, 55, 76)-net in base 8, using

base change [i] based on digital (5, 33, 76)-net over F₃₂, using
- net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
    - a function field by SÃ©mirat [i]

(55−28, 55, 103)-Net over F₈ — Digital

Digital (27, 55, 103)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−28, 55, 3040)-Net in Base 8 — Upper bound on s

There is no (27, 55, 3041)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 46 789685 499321 436814 596410 581081 070496 601284 349232 > 8⁵⁵ [i]