MinT - Best Known (109−43, 109, s)-Nets in Base 8

Best Known (109−43, 109, s)-Nets in Base 8

(109−43, 109, 354)-Net over F₈ — Constructive and digital

Digital (66, 109, 354)-net over F₈, using

9 times m-reduction [i] based on digital (66, 118, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 59, 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]

(109−43, 109, 384)-Net in Base 8 — Constructive

(66, 109, 384)-net in base 8, using

1 times m-reduction [i] based on (66, 110, 384)-net in base 8, using
- trace code for nets [i] based on (11, 55, 192)-net in base 64, using
  - 1 times m-reduction [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]

(109−43, 109, 530)-Net over F₈ — Digital

Digital (66, 109, 530)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−43, 109, 54670)-Net in Base 8 — Upper bound on s

There is no (66, 109, 54671)-net in base 8, because

1 times m-reduction [i] would yield (66, 108, 54671)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 180667 171776 120619 858058 954782 523294 160843 495131 801453 532842 857876 390678 232599 920753 017138 983568 > 8¹⁰⁸ [i]