MinT - Best Known (48, 48+57, s)-Nets in Base 9

Best Known (48, 48+57, s)-Nets in Base 9

(48, 48+57, 102)-Net over F₉ — Constructive and digital

Digital (48, 105, 102)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 31, 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, 74, 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]

(48, 48+57, 163)-Net over F₉ — Digital

Digital (48, 105, 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+57, 4928)-Net in Base 9 — Upper bound on s

There is no (48, 105, 4929)-net in base 9, because

1 times m-reduction [i] would yield (48, 104, 4929)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1743 618092 316960 634086 121519 956373 796185 454753 878980 186259 120903 163911 476492 690738 177710 203768 090465 > 9¹⁰⁴ [i]