MinT - Best Known (67, 67+43, s)-Nets in Base 8

Best Known (67, 67+43, s)-Nets in Base 8

(67, 67+43, 354)-Net over F₈ — Constructive and digital

Digital (67, 110, 354)-net over F₈, using

10 times m-reduction [i] based on digital (67, 120, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 60, 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]

(67, 67+43, 384)-Net in Base 8 — Constructive

(67, 110, 384)-net in base 8, using

2 times m-reduction [i] based on (67, 112, 384)-net in base 8, using
- trace code for nets [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(67, 67+43, 558)-Net over F₈ — Digital

Digital (67, 110, 558)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+43, 60362)-Net in Base 8 — Upper bound on s

There is no (67, 110, 60363)-net in base 8, because

1 times m-reduction [i] would yield (67, 109, 60363)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 273 442410 395476 908921 019630 786123 801620 317054 153066 964007 073686 230107 570791 100281 936104 543780 106448 > 8¹⁰⁹ [i]