MinT - Best Known (89−45, 89, s)-Nets in Base 9

Best Known (89−45, 89, s)-Nets in Base 9

(89−45, 89, 106)-Net over F₉ — Constructive and digital

Digital (44, 89, 106)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 27, 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 (17, 62, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(89−45, 89, 172)-Net over F₉ — Digital

Digital (44, 89, 172)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−45, 89, 7412)-Net in Base 9 — Upper bound on s

There is no (44, 89, 7413)-net in base 9, because

1 times m-reduction [i] would yield (44, 88, 7413)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 942181 237916 047856 404456 671802 116016 971593 004450 146095 392962 684760 984075 619734 399153 > 9⁸⁸ [i]