MinT - Best Known (139−52, 139, s)-Nets in Base 8

Best Known (139−52, 139, s)-Nets in Base 8

(139−52, 139, 354)-Net over F₈ — Constructive and digital

Digital (87, 139, 354)-net over F₈, using

21 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]

(139−52, 139, 432)-Net in Base 8 — Constructive

(87, 139, 432)-net in base 8, using

t-expansion [i] based on (85, 139, 432)-net in base 8, using
- 1 times m-reduction [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]

(139−52, 139, 837)-Net over F₈ — Digital

Digital (87, 139, 837)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−52, 139, 101434)-Net in Base 8 — Upper bound on s

There is no (87, 139, 101435)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338546 565685 418691 820465 066347 361437 977956 334429 062311 904065 157046 909907 362013 857978 721316 841699 406501 558699 849011 725385 491387 > 8¹³⁹ [i]