MinT - Best Known (54−29, 54, s)-Nets in Base 9

Best Known (54−29, 54, s)-Nets in Base 9

(54−29, 54, 78)-Net over F₉ — Constructive and digital

Digital (25, 54, 78)-net over F₉, using

t-expansion [i] based on digital (22, 54, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(54−29, 54, 82)-Net in Base 9 — Constructive

(25, 54, 82)-net in base 9, using

base change [i] based on digital (7, 36, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(54−29, 54, 96)-Net over F₉ — Digital

Digital (25, 54, 96)-net over F₉, using

net from sequence [i] based on digital (25, 95)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 25 and N(F) ≥ 96, using
  - a curve from the manYPoints database [i]

(54−29, 54, 3088)-Net in Base 9 — Upper bound on s

There is no (25, 54, 3089)-net in base 9, because

1 times m-reduction [i] would yield (25, 53, 3089)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 377 142852 028131 228203 603913 242412 767875 972972 955697 > 9⁵³ [i]