MinT - Best Known (146−85, 146, s)-Nets in Base 9

Best Known (146−85, 146, s)-Nets in Base 9

(146−85, 146, 98)-Net over F₉ — Constructive and digital

Digital (61, 146, 98)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 48, 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 (13, 98, 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]

(146−85, 146, 192)-Net over F₉ — Digital

Digital (61, 146, 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]

(146−85, 146, 4039)-Net in Base 9 — Upper bound on s

There is no (61, 146, 4040)-net in base 9, because

1 times m-reduction [i] would yield (61, 145, 4040)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 321642 184927 870085 205889 038230 357716 966232 778996 003596 820300 036817 934764 970900 826637 686309 146450 551283 192743 024321 091268 690538 016672 207745 > 9¹⁴⁵ [i]