MinT - Best Known (50, 50+91, s)-Nets in Base 9

Best Known (50, 50+91, s)-Nets in Base 9

(50, 50+91, 81)-Net over F₉ — Constructive and digital

Digital (50, 141, 81)-net over F₉, using

t-expansion [i] based on digital (32, 141, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(50, 50+91, 182)-Net over F₉ — Digital

Digital (50, 141, 182)-net over F₉, using

net from sequence [i] based on digital (50, 181)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
  - a curve from the manYPoints database [i]

(50, 50+91, 2022)-Net in Base 9 — Upper bound on s

There is no (50, 141, 2023)-net in base 9, because

1 times m-reduction [i] would yield (50, 140, 2023)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39 362732 239177 063289 292878 565720 835296 858543 542669 313477 731754 792067 720837 927683 186555 740577 891983 998120 187291 502100 581104 425980 360409 > 9¹⁴⁰ [i]