MinT - Best Known (57, 57+85, s)-Nets in Base 9

Best Known (57, 57+85, s)-Nets in Base 9

(57, 57+85, 84)-Net over F₉ — Constructive and digital

Digital (57, 142, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 44, 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, 98, 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]

(57, 57+85, 88)-Net in Base 9 — Constructive

(57, 142, 88)-net in base 9, using

2 times m-reduction [i] based on (57, 144, 88)-net in base 9, using
- base change [i] based on digital (9, 96, 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]

(57, 57+85, 182)-Net over F₉ — Digital

Digital (57, 142, 182)-net over F₉, using

t-expansion [i] based on digital (50, 142, 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]

(57, 57+85, 3272)-Net in Base 9 — Upper bound on s

There is no (57, 142, 3273)-net in base 9, because

1 times m-reduction [i] would yield (57, 141, 3273)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 356 821566 027745 390456 153197 347138 780355 609834 387739 984507 353454 577858 088686 127950 824161 388094 306214 689322 197769 684411 292126 417197 176337 > 9¹⁴¹ [i]