MinT - Best Known (90, 90+57, s)-Nets in Base 8

Best Known (90, 90+57, s)-Nets in Base 8

(90, 90+57, 354)-Net over F₈ — Constructive and digital

Digital (90, 147, 354)-net over F₈, using

19 times m-reduction [i] based on digital (90, 166, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 83, 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+57, 432)-Net in Base 8 — Constructive

(90, 147, 432)-net in base 8, using

1 times m-reduction [i] based on (90, 148, 432)-net in base 8, using
- trace code for nets [i] based on (16, 74, 216)-net in base 64, using
  - 3 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(90, 90+57, 731)-Net over F₈ — Digital

Digital (90, 147, 731)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(90, 90+57, 82562)-Net in Base 8 — Upper bound on s

There is no (90, 147, 82563)-net in base 8, because

1 times m-reduction [i] would yield (90, 146, 82563)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 709862 698691 609683 335139 637980 825435 854866 954870 920148 662660 290744 843184 039603 499751 428687 005348 670512 743186 921076 222143 695088 586592 > 8¹⁴⁶ [i]