MinT - Best Known (139−89, 139, s)-Nets in Base 9

Best Known (139−89, 139, s)-Nets in Base 9

(139−89, 139, 81)-Net over F₉ — Constructive and digital

Digital (50, 139, 81)-net over F₉, using

t-expansion [i] based on digital (32, 139, 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]

(139−89, 139, 182)-Net over F₉ — Digital

Digital (50, 139, 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]

(139−89, 139, 2094)-Net in Base 9 — Upper bound on s

There is no (50, 139, 2095)-net in base 9, because

1 times m-reduction [i] would yield (50, 138, 2095)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 486692 467838 902980 866283 593806 758111 974145 622597 812279 659634 022000 856224 896762 768301 116988 782612 009462 385910 397709 153112 139770 352801 > 9¹³⁸ [i]