MinT - Best Known (168−67, 168, s)-Nets in Base 8

Best Known (168−67, 168, s)-Nets in Base 8

(168−67, 168, 354)-Net over F₈ — Constructive and digital

Digital (101, 168, 354)-net over F₈, using

t-expansion [i] based on digital (93, 168, 354)-net over F₈, using
- 4 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(168−67, 168, 432)-Net in Base 8 — Constructive

(101, 168, 432)-net in base 8, using

trace code for nets [i] based on (17, 84, 216)-net in base 64, using
- base change [i] based on digital (5, 72, 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]

(168−67, 168, 714)-Net over F₈ — Digital

Digital (101, 168, 714)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−67, 168, 69873)-Net in Base 8 — Upper bound on s

There is no (101, 168, 69874)-net in base 8, because

1 times m-reduction [i] would yield (101, 167, 69874)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 549562 143469 704293 259126 773919 537939 716016 676551 716982 375567 672183 380218 642400 357956 609952 482450 524034 137155 390369 310978 994880 962473 953605 834424 610195 > 8¹⁶⁷ [i]