MinT - Best Known (81−72, 81, s)-Nets in Base 9

Best Known (81−72, 81, s)-Nets in Base 9

(81−72, 81, 40)-Net over F₉ — Constructive and digital

Digital (9, 81, 40)-net over F₉, using

t-expansion [i] based on digital (8, 81, 40)-net over F₉, using
- net from sequence [i] based on digital (8, 39)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
    - a function field by SÃ©mirat [i]

(81−72, 81, 48)-Net over F₉ — Digital

Digital (9, 81, 48)-net over F₉, using

net from sequence [i] based on digital (9, 47)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 9 and N(F) ≥ 48, using
  - a curve from the manYPoints database [i]

(81−72, 81, 89)-Net in Base 9 — Upper bound on s

There is no (9, 81, 90)-net in base 9, because

1 times m-reduction [i] would yield (9, 80, 90)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9⁸⁰, 90, S₉, 71), but
  - the linear programming bound shows that MÂ â‰¥ 825617 125614 748653 438025 314237 742758 149006 797066 212440 636337 770986 951542 029890 113291 / 30 000971 > 9⁸⁰ [i]