MinT - Best Known (51, 51+39, s)-Nets in Base 5

Best Known (51, 51+39, s)-Nets in Base 5

(51, 51+39, 132)-Net over F₅ — Constructive and digital

Digital (51, 90, 132)-net over F₅, using

4 times m-reduction [i] based on digital (51, 94, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 47, 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]

(51, 51+39, 158)-Net over F₅ — Digital

Digital (51, 90, 158)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+39, 3712)-Net in Base 5 — Upper bound on s

There is no (51, 90, 3713)-net in base 5, because

1 times m-reduction [i] would yield (51, 89, 3713)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 161 912197 861544 368994 301611 974582 133762 046756 898007 288084 996925 > 5⁸⁹ [i]