MinT - Best Known (139−107, 139, s)-Nets in Base 9

Best Known (139−107, 139, s)-Nets in Base 9

(139−107, 139, 81)-Net over F₉ — Constructive and digital

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]

(139−107, 139, 120)-Net over F₉ — Digital

Digital (32, 139, 120)-net over F₉, using

t-expansion [i] based on digital (31, 139, 120)-net over F₉, using
- net from sequence [i] based on digital (31, 119)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 31 and N(F) ≥ 120, using
    - a curve from the manYPoints database [i]

(139−107, 139, 753)-Net in Base 9 — Upper bound on s

There is no (32, 139, 754)-net in base 9, because

1 times m-reduction [i] would yield (32, 138, 754)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 497914 062402 736838 674057 291357 595632 808164 569452 993751 420820 113234 514136 611754 397400 341793 534467 405749 620844 545602 181514 173612 586577 > 9¹³⁸ [i]