MinT - Best Known (90, 90+51, s)-Nets in Base 8

Best Known (90, 90+51, s)-Nets in Base 8

(90, 90+51, 363)-Net over F₈ — Constructive and digital

Digital (90, 141, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 25, 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 (65, 116, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 58, 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]

(90, 90+51, 576)-Net in Base 8 — Constructive

(90, 141, 576)-net in base 8, using

8¹ times duplication [i] based on (89, 140, 576)-net in base 8, using
- trace code for nets [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]

(90, 90+51, 1005)-Net over F₈ — Digital

Digital (90, 141, 1005)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(90, 90+51, 165878)-Net in Base 8 — Upper bound on s

There is no (90, 141, 165879)-net in base 8, because

1 times m-reduction [i] would yield (90, 140, 165879)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 707756 846962 876707 857060 147528 095486 251861 396763 482330 243655 161586 117714 972881 572978 564179 323542 392725 979874 356359 575255 888088 > 8¹⁴⁰ [i]