MinT - Best Known (134−47, 134, s)-Nets in Base 8

Best Known (134−47, 134, s)-Nets in Base 8

(134−47, 134, 378)-Net over F₈ — Constructive and digital

Digital (87, 134, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (61, 108, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 54, 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]

(134−47, 134, 576)-Net in Base 8 — Constructive

(87, 134, 576)-net in base 8, using

2 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]

(134−47, 134, 1122)-Net over F₈ — Digital

Digital (87, 134, 1122)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(134−47, 134, 224670)-Net in Base 8 — Upper bound on s

There is no (87, 134, 224671)-net in base 8, because

1 times m-reduction [i] would yield (87, 133, 224671)-net in base 8, but
- 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]