MinT - Best Known (140−93, 140, s)-Nets in Base 9

Best Known (140−93, 140, s)-Nets in Base 9

(140−93, 140, 81)-Net over F₉ — Constructive and digital

Digital (47, 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]

(140−93, 140, 162)-Net over F₉ — Digital

Digital (47, 140, 162)-net over F₉, using

t-expansion [i] based on digital (46, 140, 162)-net over F₉, using
- net from sequence [i] based on digital (46, 161)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 46 and N(F) ≥ 162, using
    - a curve from the manYPoints database [i]

(140−93, 140, 1692)-Net in Base 9 — Upper bound on s

There is no (47, 140, 1693)-net in base 9, because

1 times m-reduction [i] would yield (47, 139, 1693)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 454777 583474 666296 974754 742013 505784 816151 498837 431198 659435 986799 145964 490944 853095 464816 806043 739105 530715 924440 380968 430784 407025 > 9¹³⁹ [i]