MinT - Best Known (52, 137, s)-Nets in Base 9

Best Known (52, 137, s)-Nets in Base 9

(52, 137, 81)-Net over F₉ — Constructive and digital

Digital (52, 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]

(52, 137, 182)-Net over F₉ — Digital

Digital (52, 137, 182)-net over F₉, using

t-expansion [i] based on 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]

(52, 137, 2513)-Net in Base 9 — Upper bound on s

There is no (52, 137, 2514)-net in base 9, because

1 times m-reduction [i] would yield (52, 136, 2514)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6064 207099 213384 676073 291544 238458 931950 732214 726569 056604 637643 887248 255559 389149 522981 021518 259526 486899 125960 603365 290104 177505 > 9¹³⁶ [i]