MinT - Best Known (41, 138, s)-Nets in Base 9

Best Known (41, 138, s)-Nets in Base 9

(41, 138, 81)-Net over F₉ — Constructive and digital

Digital (41, 138, 81)-net over F₉, using

t-expansion [i] based on digital (32, 138, 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]

(41, 138, 140)-Net over F₉ — Digital

Digital (41, 138, 140)-net over F₉, using

t-expansion [i] based on digital (39, 138, 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]

(41, 138, 1210)-Net in Base 9 — Upper bound on s

There is no (41, 138, 1211)-net in base 9, because

1 times m-reduction [i] would yield (41, 137, 1211)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 55416 412556 531308 043475 816048 121136 360624 048639 668076 367602 466248 492735 140230 060372 745997 825303 585502 347225 058554 601831 482053 188737 > 9¹³⁷ [i]