MinT - Best Known (144−81, 144, s)-Nets in Base 9

Best Known (144−81, 144, s)-Nets in Base 9

(144−81, 144, 108)-Net over F₉ — Constructive and digital

Digital (63, 144, 108)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 46, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (17, 98, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(144−81, 144, 192)-Net over F₉ — Digital

Digital (63, 144, 192)-net over F₉, using

t-expansion [i] based on digital (61, 144, 192)-net over F₉, using
- net from sequence [i] based on digital (61, 191)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 61 and N(F) ≥ 192, using
    - Drinfeld modules of rank 1 [i]

(144−81, 144, 5058)-Net in Base 9 — Upper bound on s

There is no (63, 144, 5059)-net in base 9, because

1 times m-reduction [i] would yield (63, 143, 5059)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 28730 900093 628345 562666 073702 724892 859166 538148 670770 910130 065187 631817 347861 917396 996000 416685 176691 233772 579102 617980 936275 713184 409025 > 9¹⁴³ [i]