MinT - Best Known (51, 51+89, s)-Nets in Base 9

Best Known (51, 51+89, s)-Nets in Base 9

(51, 51+89, 81)-Net over F₉ — Constructive and digital

Digital (51, 140, 81)-net over F₉, using

t-expansion [i] based on digital (32, 140, 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, 51+89, 182)-Net over F₉ — Digital

Digital (51, 140, 182)-net over F₉, using

t-expansion [i] based on digital (50, 140, 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, 51+89, 2203)-Net in Base 9 — Upper bound on s

There is no (51, 140, 2204)-net in base 9, because

1 times m-reduction [i] would yield (51, 139, 2204)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 410875 712956 927566 855960 133607 116913 059222 559046 196995 273133 158216 048737 924533 357523 094362 026756 336520 104282 268720 399527 721244 694913 > 9¹³⁹ [i]