MinT - Best Known (31, 31+111, s)-Nets in Base 9

Best Known (31, 31+111, s)-Nets in Base 9

(31, 31+111, 78)-Net over F₉ — Constructive and digital

Digital (31, 142, 78)-net over F₉, using

t-expansion [i] based on digital (22, 142, 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+111, 120)-Net over F₉ — Digital

Digital (31, 142, 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+111, 711)-Net in Base 9 — Upper bound on s

There is no (31, 142, 712)-net in base 9, because

1 times m-reduction [i] would yield (31, 141, 712)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 353 808418 130141 846107 730226 430923 935012 752784 268922 230385 295393 214236 243039 092690 007563 508691 105858 580615 930855 871903 593284 018837 005505 > 9¹⁴¹ [i]