MinT - Best Known (133−79, 133, s)-Nets in Base 9

Best Known (133−79, 133, s)-Nets in Base 9

(133−79, 133, 84)-Net over F₉ — Constructive and digital

Digital (54, 133, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 41, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 92, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(133−79, 133, 88)-Net in Base 9 — Constructive

(54, 133, 88)-net in base 9, using

2 times m-reduction [i] based on (54, 135, 88)-net in base 9, using
- base change [i] based on digital (9, 90, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(133−79, 133, 182)-Net over F₉ — Digital

Digital (54, 133, 182)-net over F₉, using

t-expansion [i] based on digital (50, 133, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(133−79, 133, 3242)-Net in Base 9 — Upper bound on s

There is no (54, 133, 3243)-net in base 9, because

1 times m-reduction [i] would yield (54, 132, 3243)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 915856 602550 076121 575353 653826 987190 581692 522807 115026 966890 407372 333074 197739 554236 533777 862743 704027 265606 028128 078686 597865 > 9¹³² [i]