MinT - Best Known (69−39, 69, s)-Nets in Base 9

Best Known (69−39, 69, s)-Nets in Base 9

(69−39, 69, 78)-Net over F₉ — Constructive and digital

Digital (30, 69, 78)-net over F₉, using

t-expansion [i] based on digital (22, 69, 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]

(69−39, 69, 82)-Net in Base 9 — Constructive

(30, 69, 82)-net in base 9, using

base change [i] based on digital (7, 46, 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]

(69−39, 69, 110)-Net over F₉ — Digital

Digital (30, 69, 110)-net over F₉, using

t-expansion [i] based on digital (26, 69, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(69−39, 69, 2566)-Net in Base 9 — Upper bound on s

There is no (30, 69, 2567)-net in base 9, because

1 times m-reduction [i] would yield (30, 68, 2567)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 77471 559382 826020 468441 444159 719007 173081 624273 721422 107780 008233 > 9⁶⁸ [i]