MinT - Best Known (90−39, 90, s)-Nets in Base 8

Best Known (90−39, 90, s)-Nets in Base 8

(90−39, 90, 256)-Net over F₈ — Constructive and digital

Digital (51, 90, 256)-net over F₈, using

2 times m-reduction [i] based on digital (51, 92, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 46, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(90−39, 90, 322)-Net over F₈ — Digital

Digital (51, 90, 322)-net over F₈, using

trace code for nets [i] based on digital (6, 45, 161)-net over F₆₄, using
- net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
    - a curve from the manYPoints database [i]

(90−39, 90, 19236)-Net in Base 8 — Upper bound on s

There is no (51, 90, 19237)-net in base 8, because

1 times m-reduction [i] would yield (51, 89, 19237)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 237 355872 472246 671922 780980 432997 418284 651831 623282 951340 435820 046659 382675 900232 > 8⁸⁹ [i]