MinT - Best Known (51, 142, s)-Nets in Base 9

Best Known (51, 142, s)-Nets in Base 9

(51, 142, 81)-Net over F₉ — Constructive and digital

Digital (51, 142, 81)-net over F₉, using

t-expansion [i] based on digital (32, 142, 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, 142, 182)-Net over F₉ — Digital

Digital (51, 142, 182)-net over F₉, using

t-expansion [i] based on digital (50, 142, 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, 142, 2125)-Net in Base 9 — Upper bound on s

There is no (51, 142, 2126)-net in base 9, because

1 times m-reduction [i] would yield (51, 141, 2126)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 357 235101 612315 845036 498405 572579 063683 168986 842470 194489 533282 600204 218195 252662 778162 735661 553155 825925 496471 245254 976283 646309 556529 > 9¹⁴¹ [i]