MinT - Best Known (145−83, 145, s)-Nets in Base 9

Best Known (145−83, 145, s)-Nets in Base 9

(145−83, 145, 104)-Net over F₉ — Constructive and digital

Digital (62, 145, 104)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 49, 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]
2. digital (13, 96, 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]

(145−83, 145, 192)-Net over F₉ — Digital

Digital (62, 145, 192)-net over F₉, using

t-expansion [i] based on digital (61, 145, 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]

(145−83, 145, 4507)-Net in Base 9 — Upper bound on s

There is no (62, 145, 4508)-net in base 9, because

1 times m-reduction [i] would yield (62, 144, 4508)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 259660 451711 556432 593328 421818 053045 717889 664555 377114 501274 326457 765287 109317 363625 650443 717659 277312 769736 944760 027862 164308 809344 462305 > 9¹⁴⁴ [i]