MinT - Best Known (89, 89+45, s)-Nets in Base 8

Best Known (89, 89+45, s)-Nets in Base 8

(89, 89+45, 389)-Net over F₈ — Constructive and digital

Digital (89, 134, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 30, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [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 (59, 104, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 52, 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]

(89, 89+45, 576)-Net in Base 8 — Constructive

(89, 134, 576)-net in base 8, using

6 times m-reduction [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]

(89, 89+45, 1409)-Net over F₈ — Digital

Digital (89, 134, 1409)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+45, 372673)-Net in Base 8 — Upper bound on s

There is no (89, 134, 372674)-net in base 8, because

1 times m-reduction [i] would yield (89, 133, 372674)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 291160 208922 408147 408324 418768 060615 364179 702829 297863 053598 758989 659099 346625 104985 140137 248460 031513 590685 665830 652832 > 8¹³³ [i]