MinT - Best Known (111−47, 111, s)-Nets in Base 9

Best Known (111−47, 111, s)-Nets in Base 9

(111−47, 111, 344)-Net over F₉ — Constructive and digital

Digital (64, 111, 344)-net over F₉, using

3 times m-reduction [i] based on digital (64, 114, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 57, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(111−47, 111, 452)-Net over F₉ — Digital

Digital (64, 111, 452)-net over F₉, using

1 times m-reduction [i] based on digital (64, 112, 452)-net over F₉, using
- trace code for nets [i] based on digital (8, 56, 226)-net over F₈₁, using
  - net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
      - a curve from the manYPoints database [i]

(111−47, 111, 43151)-Net in Base 9 — Upper bound on s

There is no (64, 111, 43152)-net in base 9, because

1 times m-reduction [i] would yield (64, 110, 43152)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 926 419281 283685 753057 194035 074394 856928 567567 010305 385121 159387 240112 406732 513991 239148 037792 679992 127873 > 9¹¹⁰ [i]