MinT - Best Known (109−78, 109, s)-Nets in Base 9

Best Known (109−78, 109, s)-Nets in Base 9

(109−78, 109, 78)-Net over F₉ — Constructive and digital

Digital (31, 109, 78)-net over F₉, using

t-expansion [i] based on digital (22, 109, 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]

(109−78, 109, 120)-Net over F₉ — Digital

Digital (31, 109, 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]

(109−78, 109, 870)-Net in Base 9 — Upper bound on s

There is no (31, 109, 871)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 106 206124 718307 327656 266576 225400 516161 055055 415612 410592 428799 118224 852677 815426 800860 949825 028462 347401 > 9¹⁰⁹ [i]