MinT - Best Known (141−81, 141, s)-Nets in Base 9

Best Known (141−81, 141, s)-Nets in Base 9

(141−81, 141, 102)-Net over F₉ — Constructive and digital

Digital (60, 141, 102)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 43, 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, 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]

(141−81, 141, 190)-Net over F₉ — Digital

Digital (60, 141, 190)-net over F₉, using

net from sequence [i] based on digital (60, 189)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 60 and N(F) ≥ 190, using
  - Drinfeld modules of rank 1 [i]

(141−81, 141, 4286)-Net in Base 9 — Upper bound on s

There is no (60, 141, 4287)-net in base 9, because

1 times m-reduction [i] would yield (60, 140, 4287)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39 529892 569414 720655 274098 470422 307769 500852 731970 470688 286458 376429 608772 970289 770892 485773 641437 583910 986887 918254 590468 740029 599425 > 9¹⁴⁰ [i]