MinT - Best Known (137−87, 137, s)-Nets in Base 9

Best Known (137−87, 137, s)-Nets in Base 9

(137−87, 137, 81)-Net over F₉ — Constructive and digital

Digital (50, 137, 81)-net over F₉, using

t-expansion [i] based on digital (32, 137, 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]

(137−87, 137, 182)-Net over F₉ — Digital

Digital (50, 137, 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]

(137−87, 137, 2173)-Net in Base 9 — Upper bound on s

There is no (50, 137, 2174)-net in base 9, because

1 times m-reduction [i] would yield (50, 136, 2174)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6002 026149 984884 245015 960825 970964 418769 706448 048812 051587 218565 308448 222655 941972 498875 154188 051888 253049 726539 866610 200077 728529 > 9¹³⁶ [i]