MinT - Best Known (111−99, 111, s)-Nets in Base 9

Best Known (111−99, 111, s)-Nets in Base 9

(111−99, 111, 40)-Net over F₉ — Constructive and digital

Digital (12, 111, 40)-net over F₉, using

t-expansion [i] based on digital (8, 111, 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]

(111−99, 111, 56)-Net over F₉ — Digital

Digital (12, 111, 56)-net over F₉, using

net from sequence [i] based on digital (12, 55)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 12 and N(F) ≥ 56, using
  - a curve from the manYPoints database [i]

(111−99, 111, 113)-Net in Base 9 — Upper bound on s

There is no (12, 111, 114)-net in base 9, because

12 times m-reduction [i] would yield (12, 99, 114)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9⁹⁹, 114, S₉, 87), but
  - the linear programming bound shows that MÂ â‰¥ 183 712452 224151 496359 730756 773016 045194 033482 014231 382801 426138 213217 418543 740185 901581 517780 371424 905718 474237 / 4987 502939 947913 > 9⁹⁹ [i]