MinT - Best Known (57, 57+81, s)-Nets in Base 9

Best Known (57, 57+81, s)-Nets in Base 9

(57, 57+81, 94)-Net over F₉ — Constructive and digital

Digital (57, 138, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 44, 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, 94, 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+81, 96)-Net in Base 9 — Constructive

(57, 138, 96)-net in base 9, using

base change [i] based on digital (11, 92, 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]

(57, 57+81, 182)-Net over F₉ — Digital

Digital (57, 138, 182)-net over F₉, using

t-expansion [i] based on digital (50, 138, 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+81, 3631)-Net in Base 9 — Upper bound on s

There is no (57, 138, 3632)-net in base 9, because

1 times m-reduction [i] would yield (57, 137, 3632)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54243 667618 924204 399630 762601 095315 243867 895257 894848 630481 074651 393008 087459 785356 546150 110297 801233 527476 436276 259347 929063 228417 > 9¹³⁷ [i]