MinT - Best Known (81, 81+63, s)-Nets in Base 8

Best Known (81, 81+63, s)-Nets in Base 8

(81, 81+63, 354)-Net over F₈ — Constructive and digital

Digital (81, 144, 354)-net over F₈, using

4 times m-reduction [i] based on digital (81, 148, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 74, 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]

(81, 81+63, 418)-Net over F₈ — Digital

Digital (81, 144, 418)-net over F₈, using

trace code for nets [i] based on digital (9, 72, 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]

(81, 81+63, 25970)-Net in Base 8 — Upper bound on s

There is no (81, 144, 25971)-net in base 8, because

1 times m-reduction [i] would yield (81, 143, 25971)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1387 496287 889354 677952 690811 450414 114354 828143 934954 657401 443565 437469 772418 078955 739814 792278 125114 930081 822636 832609 709887 672672 > 8¹⁴³ [i]