MinT - Best Known (75−31, 75, s)-Nets in Base 5

Best Known (75−31, 75, s)-Nets in Base 5

(75−31, 75, 132)-Net over F₅ — Constructive and digital

Digital (44, 75, 132)-net over F₅, using

5 times m-reduction [i] based on digital (44, 80, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 40, 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]

(75−31, 75, 162)-Net over F₅ — Digital

Digital (44, 75, 162)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−31, 75, 4496)-Net in Base 5 — Upper bound on s

There is no (44, 75, 4497)-net in base 5, because

1 times m-reduction [i] would yield (44, 74, 4497)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 5294 001901 510717 123011 785213 012278 254879 811904 539341 > 5⁷⁴ [i]