MinT - Best Known (165−78, 165, s)-Nets in Base 8

Best Known (165−78, 165, s)-Nets in Base 8

(165−78, 165, 208)-Net over F₈ — Constructive and digital

Digital (87, 165, 208)-net over F₈, using

3 times m-reduction [i] based on digital (87, 168, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 84, 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]

(165−78, 165, 225)-Net in Base 8 — Constructive

(87, 165, 225)-net in base 8, using

t-expansion [i] based on (83, 165, 225)-net in base 8, using
- 7 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 225)-net over F₁₆, using
    - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
        a function field by GarcÃa and Quoos [i]

(165−78, 165, 334)-Net over F₈ — Digital

Digital (87, 165, 334)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−78, 165, 14533)-Net in Base 8 — Upper bound on s

There is no (87, 165, 14534)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102531 064625 885956 877567 071027 776294 299055 509690 631327 438019 650489 121298 906128 718300 156548 157649 688217 878975 810917 863069 217876 904865 538232 513716 032160 > 8¹⁶⁵ [i]