MinT - Best Known (85, 138, s)-Nets in Base 8

Best Known (85, 138, s)-Nets in Base 8

(85, 138, 354)-Net over F₈ — Constructive and digital

Digital (85, 138, 354)-net over F₈, using

18 times m-reduction [i] based on digital (85, 156, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 78, 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]

(85, 138, 432)-Net in Base 8 — Constructive

(85, 138, 432)-net in base 8, using

2 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]

(85, 138, 729)-Net over F₈ — Digital

Digital (85, 138, 729)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 138, 86437)-Net in Base 8 — Upper bound on s

There is no (85, 138, 86438)-net in base 8, because

1 times m-reduction [i] would yield (85, 137, 86438)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5288 959772 283039 068057 045251 652927 732982 681466 488990 966886 371047 360824 523893 452960 757261 186494 669476 146410 764768 638637 236024 > 8¹³⁷ [i]