MinT - Best Known (87, 87+55, s)-Nets in Base 8

Best Known (87, 87+55, s)-Nets in Base 8

(87, 87+55, 354)-Net over F₈ — Constructive and digital

Digital (87, 142, 354)-net over F₈, using

18 times m-reduction [i] based on digital (87, 160, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 80, 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+55, 432)-Net in Base 8 — Constructive

(87, 142, 432)-net in base 8, using

8² times duplication [i] based on (85, 140, 432)-net in base 8, using
- trace code for nets [i] based on (15, 70, 216)-net in base 64, using
  - base change [i] based on digital (5, 60, 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]

(87, 87+55, 715)-Net over F₈ — Digital

Digital (87, 142, 715)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(87, 87+55, 81162)-Net in Base 8 — Upper bound on s

There is no (87, 142, 81163)-net in base 8, because

1 times m-reduction [i] would yield (87, 141, 81163)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 665452 654187 422392 964108 617107 059553 024165 948976 295059 847216 607656 733934 818091 777020 446862 607007 771311 200067 067866 223452 037760 > 8¹⁴¹ [i]