MinT - Best Known (57, 57+45, s)-Nets in Base 5

Best Known (57, 57+45, s)-Nets in Base 5

(57, 57+45, 132)-Net over F₅ — Constructive and digital

Digital (57, 102, 132)-net over F₅, using

4 times m-reduction [i] based on digital (57, 106, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 53, 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, 57+45, 164)-Net over F₅ — Digital

Digital (57, 102, 164)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+45, 3645)-Net in Base 5 — Upper bound on s

There is no (57, 102, 3646)-net in base 5, because

1 times m-reduction [i] would yield (57, 101, 3646)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 39472 175654 652372 908346 883206 943925 284652 210375 436198 877549 176343 833025 > 5¹⁰¹ [i]