MinT - Best Known (54, 54+22, s)-Nets in Base 8

Best Known (54, 54+22, s)-Nets in Base 8

(54, 54+22, 388)-Net over F₈ — Constructive and digital

Digital (54, 76, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 18, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (36, 58, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 29, 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+22, 576)-Net in Base 8 — Constructive

(54, 76, 576)-net in base 8, using

2 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+22, 2310)-Net over F₈ — Digital

Digital (54, 76, 2310)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+22, 1217431)-Net in Base 8 — Upper bound on s

There is no (54, 76, 1217432)-net in base 8, because

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]