MinT - Best Known (123−71, 123, s)-Nets in Base 9

Best Known (123−71, 123, s)-Nets in Base 9

(123−71, 123, 94)-Net over F₉ — Constructive and digital

Digital (52, 123, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 39, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (13, 84, 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]

(123−71, 123, 96)-Net in Base 9 — Constructive

(52, 123, 96)-net in base 9, using

base change [i] based on digital (11, 82, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(123−71, 123, 182)-Net over F₉ — Digital

Digital (52, 123, 182)-net over F₉, using

t-expansion [i] based on digital (50, 123, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(123−71, 123, 3663)-Net in Base 9 — Upper bound on s

There is no (52, 123, 3664)-net in base 9, because

1 times m-reduction [i] would yield (52, 122, 3664)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 263 405909 566723 481029 898946 997208 688913 965192 855694 835852 157662 375597 572723 356069 195351 542113 426492 815153 574039 479681 > 9¹²² [i]