MinT - Best Known (89−19, 89, s)-Nets in Base 5

Best Known (89−19, 89, s)-Nets in Base 5

(89−19, 89, 416)-Net over F₅ — Constructive and digital

Digital (70, 89, 416)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (24, 33, 208)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (5, 9, 137)-net over F₅, using
      - linear code embedding found in a computer search [i]
    2. digital (15, 24, 104)-net over F₅, using
      - trace code for nets [i] based on digital (3, 12, 52)-net over F₂₅, using
        net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
        a function field by GarcÃa and Quoos [i]
2. digital (37, 56, 208)-net over F₅, using
  - trace code for nets [i] based on digital (9, 28, 104)-net over F₂₅, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
        a function field by GarcÃa and Quoos [i]

(89−19, 89, 5405)-Net over F₅ — Digital

Digital (70, 89, 5405)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−19, 89, 7080508)-Net in Base 5 — Upper bound on s

There is no (70, 89, 7080509)-net in base 5, because

1 times m-reduction [i] would yield (70, 88, 7080509)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 32 311762 769454 054991 953669 833113 031481 461001 350077 408296 041045 > 5⁸⁸ [i]