MinT - Best Known (123−53, 123, s)-Nets in Base 9

Best Known (123−53, 123, s)-Nets in Base 9

(123−53, 123, 344)-Net over F₉ — Constructive and digital

Digital (70, 123, 344)-net over F₉, using

3 times m-reduction [i] based on digital (70, 126, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 63, 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]

(123−53, 123, 452)-Net over F₉ — Digital

Digital (70, 123, 452)-net over F₉, using

1 times m-reduction [i] based on digital (70, 124, 452)-net over F₉, using
- trace code for nets [i] based on digital (8, 62, 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]

(123−53, 123, 39594)-Net in Base 9 — Upper bound on s

There is no (70, 123, 39595)-net in base 9, because

1 times m-reduction [i] would yield (70, 122, 39595)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 261 733798 090047 047031 828655 966830 886616 933783 028604 622510 699983 675462 054571 897455 451971 312709 354369 257306 829048 224113 > 9¹²² [i]