MinT - Best Known (168−81, 168, s)-Nets in Base 8

Best Known (168−81, 168, s)-Nets in Base 8

(168−81, 168, 208)-Net over F₈ — Constructive and digital

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]

(168−81, 168, 225)-Net in Base 8 — Constructive

(87, 168, 225)-net in base 8, using

t-expansion [i] based on (83, 168, 225)-net in base 8, using
- 4 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]

(168−81, 168, 312)-Net over F₈ — Digital

Digital (87, 168, 312)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−81, 168, 13251)-Net in Base 8 — Upper bound on s

There is no (87, 168, 13252)-net in base 8, because

1 times m-reduction [i] would yield (87, 167, 13252)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 554446 377613 662384 291160 624274 799270 247504 408982 804838 378087 766439 901842 720671 597829 143101 152986 176608 653049 442115 589447 888961 647654 305287 907722 280056 > 8¹⁶⁷ [i]