MinT - Best Known (48, 48+61, s)-Nets in Base 9

Best Known (48, 48+61, s)-Nets in Base 9

(48, 48+61, 96)-Net over F₉ — Constructive and digital

Digital (48, 109, 96)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 35, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 74, 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]

(48, 48+61, 163)-Net over F₉ — Digital

Digital (48, 109, 163)-net over F₉, using

net from sequence [i] based on digital (48, 162)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 48 and N(F) ≥ 163, using
  - Drinfeld modules of rank 1 [i]

(48, 48+61, 4083)-Net in Base 9 — Upper bound on s

There is no (48, 109, 4084)-net in base 9, because

1 times m-reduction [i] would yield (48, 108, 4084)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11 471679 841657 345184 883825 304639 246349 207264 804266 125957 354788 905247 507762 145821 317676 553984 080623 504961 > 9¹⁰⁸ [i]