MinT - Best Known (86−37, 86, s)-Nets in Base 8

Best Known (86−37, 86, s)-Nets in Base 8

(86−37, 86, 256)-Net over F₈ — Constructive and digital

Digital (49, 86, 256)-net over F₈, using

2 times m-reduction [i] based on digital (49, 88, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 44, 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]

(86−37, 86, 322)-Net over F₈ — Digital

Digital (49, 86, 322)-net over F₈, using

trace code for nets [i] based on digital (6, 43, 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]

(86−37, 86, 19832)-Net in Base 8 — Upper bound on s

There is no (49, 86, 19833)-net in base 8, because

1 times m-reduction [i] would yield (49, 85, 19833)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 57913 125783 219469 224132 014515 330130 372161 120508 647544 572045 427515 224134 645276 > 8⁸⁵ [i]