MinT - Best Known (126−37, 126, s)-Nets in Base 9

Best Known (126−37, 126, s)-Nets in Base 9

(126−37, 126, 760)-Net over F₉ — Constructive and digital

Digital (89, 126, 760)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (69, 106, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 53, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(126−37, 126, 3922)-Net over F₉ — Digital

Digital (89, 126, 3922)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−37, 126, 3996897)-Net in Base 9 — Upper bound on s

There is no (89, 126, 3996898)-net in base 9, because

1 times m-reduction [i] would yield (89, 125, 3996898)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 190684 304099 695209 967858 232644 536736 365499 074148 072616 261656 144216 913671 041132 210780 057130 481131 800613 816389 860772 814305 > 9¹²⁵ [i]