MinT - Best Known (58, 58+83, s)-Nets in Base 9

Best Known (58, 58+83, s)-Nets in Base 9

(58, 58+83, 94)-Net over F₉ — Constructive and digital

Digital (58, 141, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 45, 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, 96, 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]

(58, 58+83, 96)-Net in Base 9 — Constructive

(58, 141, 96)-net in base 9, using

base change [i] based on digital (11, 94, 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]

(58, 58+83, 182)-Net over F₉ — Digital

Digital (58, 141, 182)-net over F₉, using

t-expansion [i] based on digital (50, 141, 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]

(58, 58+83, 3632)-Net in Base 9 — Upper bound on s

There is no (58, 141, 3633)-net in base 9, because

1 times m-reduction [i] would yield (58, 140, 3633)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39 410762 288284 573602 191299 762217 153442 101076 293456 471595 273671 867232 409168 568558 819706 897183 044546 566783 941316 557604 213493 028530 293705 > 9¹⁴⁰ [i]