MinT - Best Known (54, 54+23, s)-Nets in Base 8

Best Known (54, 54+23, s)-Nets in Base 8

(54, 54+23, 382)-Net over F₈ — Constructive and digital

Digital (54, 77, 382)-net over F₈, using

8¹ times duplication [i] based on digital (53, 76, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 16, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (37, 60, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 30, 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+23, 576)-Net in Base 8 — Constructive

(54, 77, 576)-net in base 8, using

1 times m-reduction [i] based on (54, 78, 576)-net in base 8, using
- trace code for nets [i] based on (15, 39, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
    - base change [i] based on digital (9, 36, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(54, 54+23, 1884)-Net over F₈ — Digital

Digital (54, 77, 1884)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+23, 1217431)-Net in Base 8 — Upper bound on s

There is no (54, 77, 1217432)-net in base 8, because

1 times m-reduction [i] would yield (54, 76, 1217432)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 431 360476 911543 499397 855919 724615 358135 037889 293538 423001 384069 561890 > 8⁷⁶ [i]