MinT - Best Known (45, 45+31, s)-Nets in Base 5

Best Known (45, 45+31, s)-Nets in Base 5

(45, 45+31, 132)-Net over F₅ — Constructive and digital

Digital (45, 76, 132)-net over F₅, using

6 times m-reduction [i] based on digital (45, 82, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 41, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(45, 45+31, 172)-Net over F₅ — Digital

Digital (45, 76, 172)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+31, 5007)-Net in Base 5 — Upper bound on s

There is no (45, 76, 5008)-net in base 5, because

1 times m-reduction [i] would yield (45, 75, 5008)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 26504 694864 023175 095897 680843 900020 630430 678556 546625 > 5⁷⁵ [i]