MinT - Best Known (87, 87+46, s)-Nets in Base 8

Best Known (87, 87+46, s)-Nets in Base 8

(87, 87+46, 379)-Net over F₈ — Constructive and digital

Digital (87, 133, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 27, 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 (60, 106, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 53, 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]

(87, 87+46, 576)-Net in Base 8 — Constructive

(87, 133, 576)-net in base 8, using

3 times m-reduction [i] based on (87, 136, 576)-net in base 8, using
- trace code for nets [i] based on (19, 68, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(87, 87+46, 1198)-Net over F₈ — Digital

Digital (87, 133, 1198)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(87, 87+46, 224670)-Net in Base 8 — Upper bound on s

There is no (87, 133, 224671)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 291150 944829 709945 821724 189680 997158 342548 849321 027063 173519 679781 484505 180554 093366 856127 409080 118567 923621 947638 486592 > 8¹³³ [i]