MinT - Best Known (131−91, 131, s)-Nets in Base 9

Best Known (131−91, 131, s)-Nets in Base 9

(131−91, 131, 81)-Net over F₉ — Constructive and digital

Digital (40, 131, 81)-net over F₉, using

t-expansion [i] based on digital (32, 131, 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]

(131−91, 131, 140)-Net over F₉ — Digital

Digital (40, 131, 140)-net over F₉, using

t-expansion [i] based on digital (39, 131, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(131−91, 131, 1230)-Net in Base 9 — Upper bound on s

There is no (40, 131, 1231)-net in base 9, because

1 times m-reduction [i] would yield (40, 130, 1231)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11319 805416 387705 329253 572713 949862 240029 112270 718133 563002 726712 062220 705444 717367 306629 947904 694138 928338 942102 141148 763929 > 9¹³⁰ [i]