MinT - Best Known (42, 42+53, s)-Nets in Base 9

Best Known (42, 42+53, s)-Nets in Base 9

(42, 42+53, 92)-Net over F₉ — Constructive and digital

Digital (42, 95, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 29, 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 (13, 66, 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]

(42, 42+53, 94)-Net in Base 9 — Constructive

(42, 95, 94)-net in base 9, using

1 times m-reduction [i] based on (42, 96, 94)-net in base 9, using
- base change [i] based on digital (10, 64, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(42, 42+53, 140)-Net over F₉ — Digital

Digital (42, 95, 140)-net over F₉, using

t-expansion [i] based on digital (39, 95, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(42, 42+53, 3700)-Net in Base 9 — Upper bound on s

There is no (42, 95, 3701)-net in base 9, because

1 times m-reduction [i] would yield (42, 94, 3701)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 499928 033628 637037 852670 430129 475346 354519 547797 075813 855269 607822 649064 008543 402857 534417 > 9⁹⁴ [i]