MinT - Best Known (68, 68+39, s)-Nets in Base 8

Best Known (68, 68+39, s)-Nets in Base 8

(68, 68+39, 354)-Net over F₈ — Constructive and digital

Digital (68, 107, 354)-net over F₈, using

15 times m-reduction [i] based on digital (68, 122, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 61, 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]

(68, 68+39, 514)-Net in Base 8 — Constructive

(68, 107, 514)-net in base 8, using

1 times m-reduction [i] based on (68, 108, 514)-net in base 8, using
- base change [i] based on digital (41, 81, 514)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (41, 82, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 41, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(68, 68+39, 768)-Net over F₈ — Digital

Digital (68, 107, 768)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+39, 123699)-Net in Base 8 — Upper bound on s

There is no (68, 107, 123700)-net in base 8, because

1 times m-reduction [i] would yield (68, 106, 123700)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 534040 004607 187876 908122 053309 514797 930352 327599 860714 919130 312865 363213 282993 709880 928787 606321 > 8¹⁰⁶ [i]