MinT - Best Known (78−31, 78, s)-Nets in Base 5

Best Known (78−31, 78, s)-Nets in Base 5

(78−31, 78, 132)-Net over F₅ — Constructive and digital

Digital (47, 78, 132)-net over F₅, using

8 times m-reduction [i] based on digital (47, 86, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 43, 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]

(78−31, 78, 194)-Net over F₅ — Digital

Digital (47, 78, 194)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−31, 78, 6208)-Net in Base 5 — Upper bound on s

There is no (47, 78, 6209)-net in base 5, because

1 times m-reduction [i] would yield (47, 77, 6209)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 662196 832428 959145 722368 211518 096552 521496 541450 458765 > 5⁷⁷ [i]