MinT - Best Known (68, 109, s)-Nets in Base 8

Best Known (68, 109, s)-Nets in Base 8

(68, 109, 354)-Net over F₈ — Constructive and digital

Digital (68, 109, 354)-net over F₈, using

13 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, 109, 432)-Net in Base 8 — Constructive

(68, 109, 432)-net in base 8, using

1 times m-reduction [i] based on (68, 110, 432)-net in base 8, using
- trace code for nets [i] based on (13, 55, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (13, 56, 216)-net in base 64, using
    - base change [i] based on digital (5, 48, 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]

(68, 109, 671)-Net over F₈ — Digital

Digital (68, 109, 671)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 109, 89296)-Net in Base 8 — Upper bound on s

There is no (68, 109, 89297)-net in base 8, because

1 times m-reduction [i] would yield (68, 108, 89297)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 180262 775893 428031 362515 612347 488544 533197 187512 571170 791244 007231 119145 556200 845698 946923 644496 > 8¹⁰⁸ [i]