MinT - Best Known (54, 54+87, s)-Nets in Base 9

Best Known (54, 54+87, s)-Nets in Base 9

(54, 54+87, 81)-Net over F₉ — Constructive and digital

Digital (54, 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]

(54, 54+87, 82)-Net in Base 9 — Constructive

(54, 141, 82)-net in base 9, using

base change [i] based on digital (7, 94, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(54, 54+87, 182)-Net over F₉ — Digital

Digital (54, 141, 182)-net over F₉, using

t-expansion [i] based on 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]

(54, 54+87, 2672)-Net in Base 9 — Upper bound on s

There is no (54, 141, 2673)-net in base 9, because

1 times m-reduction [i] would yield (54, 140, 2673)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39 402204 864192 794715 459676 639547 963718 701612 098550 358411 856151 703972 535507 786429 384733 306734 236465 657910 358934 752938 314547 888475 007385 > 9¹⁴⁰ [i]