MinT - Best Known (145−45, 145, s)-Nets in Base 9

Best Known (145−45, 145, s)-Nets in Base 9

(145−45, 145, 756)-Net over F₉ — Constructive and digital

Digital (100, 145, 756)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (77, 122, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 61, 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]

(145−45, 145, 3031)-Net over F₉ — Digital

Digital (100, 145, 3031)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(145−45, 145, 1993935)-Net in Base 9 — Upper bound on s

There is no (100, 145, 1993936)-net in base 9, because

1 times m-reduction [i] would yield (100, 144, 1993936)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 257586 638267 461738 202471 557101 939516 558567 908107 230259 221039 972245 702519 092346 273545 683016 560768 978191 285038 651967 144516 163448 460203 255041 > 9¹⁴⁴ [i]