MinT - Best Known (144−83, 144, s)-Nets in Base 9

Best Known (144−83, 144, s)-Nets in Base 9

(144−83, 144, 102)-Net over F₉ — Constructive and digital

Digital (61, 144, 102)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 44, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (17, 100, 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]

(144−83, 144, 192)-Net over F₉ — Digital

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]

(144−83, 144, 4270)-Net in Base 9 — Upper bound on s

There is no (61, 144, 4271)-net in base 9, because

1 times m-reduction [i] would yield (61, 143, 4271)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 28723 373852 956032 768237 379000 351271 167514 628391 914132 012585 530789 236590 730672 810969 385687 239671 698563 605426 264843 840266 059746 301747 636281 > 9¹⁴³ [i]