MinT - Best Known (48, 48+67, s)-Nets in Base 9

Best Known (48, 48+67, s)-Nets in Base 9

(48, 48+67, 84)-Net over F₉ — Constructive and digital

Digital (48, 115, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 35, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 80, 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+67, 88)-Net in Base 9 — Constructive

(48, 115, 88)-net in base 9, using

2 times m-reduction [i] based on (48, 117, 88)-net in base 9, using
- base change [i] based on digital (9, 78, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(48, 48+67, 163)-Net over F₉ — Digital

Digital (48, 115, 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+67, 3236)-Net in Base 9 — Upper bound on s

There is no (48, 115, 3237)-net in base 9, because

1 times m-reduction [i] would yield (48, 114, 3237)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 118694 735400 027900 288152 704281 553616 422837 913849 950890 429675 962115 644694 735602 243279 504297 178369 498441 147945 > 9¹¹⁴ [i]