MinT - Best Known (128−55, 128, s)-Nets in Base 8

Best Known (128−55, 128, s)-Nets in Base 8

(128−55, 128, 354)-Net over F₈ — Constructive and digital

Digital (73, 128, 354)-net over F₈, using

4 times m-reduction [i] based on digital (73, 132, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 66, 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]

(128−55, 128, 418)-Net over F₈ — Digital

Digital (73, 128, 418)-net over F₈, using

trace code for nets [i] based on digital (9, 64, 209)-net over F₆₄, using
- net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
    - a curve from the manYPoints database [i]

(128−55, 128, 27600)-Net in Base 8 — Upper bound on s

There is no (73, 128, 27601)-net in base 8, because

1 times m-reduction [i] would yield (73, 127, 27601)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 929380 820983 972220 762418 759157 281203 627777 272986 365919 822917 871085 491383 544464 928128 939383 056301 609032 055956 826944 > 8¹²⁷ [i]