MinT - Best Known (51, 138, s)-Nets in Base 9

Best Known (51, 138, s)-Nets in Base 9

(51, 138, 81)-Net over F₉ — Constructive and digital

Digital (51, 138, 81)-net over F₉, using

t-expansion [i] based on digital (32, 138, 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]

(51, 138, 182)-Net over F₉ — Digital

Digital (51, 138, 182)-net over F₉, using

t-expansion [i] based on digital (50, 138, 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]

(51, 138, 2289)-Net in Base 9 — Upper bound on s

There is no (51, 138, 2290)-net in base 9, because

1 times m-reduction [i] would yield (51, 137, 2290)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54676 079928 396240 504597 006529 374797 886426 619804 926461 647712 101555 560586 644567 702334 939428 326663 363419 937491 374874 198736 274365 811825 > 9¹³⁷ [i]