MinT - Best Known (87, 87+32, s)-Nets in Base 8

Best Known (87, 87+32, s)-Nets in Base 8

(87, 87+32, 562)-Net over F₈ — Constructive and digital

Digital (87, 119, 562)-net over F₈, using

8¹ times duplication [i] based on digital (86, 118, 562)-net over F₈, using
- t-expansion [i] based on digital (85, 118, 562)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (22, 38, 208)-net over F₈, using
      - trace code for nets [i] based on digital (3, 19, 104)-net over F₆₄, using
        net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
    2. digital (47, 80, 354)-net over F₈, using
      - trace code for nets [i] based on digital (7, 40, 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+32, 654)-Net in Base 8 — Constructive

(87, 119, 654)-net in base 8, using

1 times m-reduction [i] based on (87, 120, 654)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (24, 40, 300)-net in base 8, using
    - trace code for nets [i] based on (4, 20, 150)-net in base 64, using
      - 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
        base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
        net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (47, 80, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 40, 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+32, 5211)-Net over F₈ — Digital

Digital (87, 119, 5211)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(87, 87+32, 5060245)-Net in Base 8 — Upper bound on s

There is no (87, 119, 5060246)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 293567 901300 953295 319475 389853 551752 944576 374617 881917 478601 991191 080299 748052 547442 002537 989087 121891 480533 > 8¹¹⁹ [i]