MinT - Best Known (54, 54+27, s)-Nets in Base 8

Best Known (54, 54+27, s)-Nets in Base 8

(54, 54+27, 363)-Net over F₈ — Constructive and digital

Digital (54, 81, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. 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]

(54, 54+27, 520)-Net in Base 8 — Constructive

(54, 81, 520)-net in base 8, using

8¹ times duplication [i] based on (53, 80, 520)-net in base 8, using
- base change [i] based on digital (33, 60, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 30, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(54, 54+27, 994)-Net over F₈ — Digital

Digital (54, 81, 994)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+27, 292262)-Net in Base 8 — Upper bound on s

There is no (54, 81, 292263)-net in base 8, because

1 times m-reduction [i] would yield (54, 80, 292263)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 766875 197705 066048 630366 416681 654658 341583 643820 221414 093968 002560 887912 > 8⁸⁰ [i]