MinT - Best Known (128−37, 128, s)-Nets in Base 9

Best Known (128−37, 128, s)-Nets in Base 9

(128−37, 128, 770)-Net over F₉ — Constructive and digital

Digital (91, 128, 770)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 22, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [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]

(128−37, 128, 4429)-Net over F₉ — Digital

Digital (91, 128, 4429)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−37, 128, 5102114)-Net in Base 9 — Upper bound on s

There is no (91, 128, 5102115)-net in base 9, because

1 times m-reduction [i] would yield (91, 127, 5102115)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15 445415 459342 172738 813394 580468 395390 627733 443445 750487 875865 712855 931880 305785 482542 119182 685737 716925 606466 678494 991665 > 9¹²⁷ [i]