MinT - Best Known (31, 31+51, s)-Nets in Base 9

Best Known (31, 31+51, s)-Nets in Base 9

(31, 31+51, 78)-Net over F₉ — Constructive and digital

Digital (31, 82, 78)-net over F₉, using

t-expansion [i] based on digital (22, 82, 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]

(31, 31+51, 120)-Net over F₉ — Digital

Digital (31, 82, 120)-net over F₉, using

net from sequence [i] based on digital (31, 119)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 31 and N(F) ≥ 120, using
  - a curve from the manYPoints database [i]

(31, 31+51, 1556)-Net in Base 9 — Upper bound on s

There is no (31, 82, 1557)-net in base 9, because

1 times m-reduction [i] would yield (31, 81, 1557)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 198830 228718 109918 439655 132862 521135 713262 100633 069231 985553 322572 199857 050985 > 9⁸¹ [i]