MinT - Best Known (89, 89+35, s)-Nets in Base 9

Best Known (89, 89+35, s)-Nets in Base 9

(89, 89+35, 772)-Net over F₉ — Constructive and digital

Digital (89, 124, 772)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 22, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]
2. digital (67, 102, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 51, 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]

(89, 89+35, 5129)-Net over F₉ — Digital

Digital (89, 124, 5129)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+35, 7195812)-Net in Base 9 — Upper bound on s

There is no (89, 124, 7195813)-net in base 9, because

1 times m-reduction [i] would yield (89, 123, 7195813)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2354 124457 758877 070340 658721 257109 812523 187908 744856 788716 047411 868238 359645 999123 347617 349371 502490 885884 116168 934825 > 9¹²³ [i]