MinT - Best Known (57−21, 57, s)-Nets in Base 5

Best Known (57−21, 57, s)-Nets in Base 5

(57−21, 57, 132)-Net over F₅ — Constructive and digital

Digital (36, 57, 132)-net over F₅, using

7 times m-reduction [i] based on digital (36, 64, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 32, 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]

(57−21, 57, 214)-Net over F₅ — Digital

Digital (36, 57, 214)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−21, 57, 9285)-Net in Base 5 — Upper bound on s

There is no (36, 57, 9286)-net in base 5, because

1 times m-reduction [i] would yield (36, 56, 9286)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1388 018765 235586 280875 852345 605538 309121 > 5⁵⁶ [i]