MinT - Best Known (56−21, 56, s)-Nets in Base 5

Best Known (56−21, 56, s)-Nets in Base 5

(56−21, 56, 132)-Net over F₅ — Constructive and digital

Digital (35, 56, 132)-net over F₅, using

6 times m-reduction [i] based on digital (35, 62, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 31, 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]

(56−21, 56, 198)-Net over F₅ — Digital

Digital (35, 56, 198)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−21, 56, 7904)-Net in Base 5 — Upper bound on s

There is no (35, 56, 7905)-net in base 5, because

1 times m-reduction [i] would yield (35, 55, 7905)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 277 778357 115455 810261 265247 624557 569785 > 5⁵⁵ [i]