MinT - Best Known (57, 57+49, s)-Nets in Base 5

Best Known (57, 57+49, s)-Nets in Base 5

(57, 57+49, 132)-Net over F₅ — Constructive and digital

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+49, 143)-Net over F₅ — Digital

Digital (57, 106, 143)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+49, 2783)-Net in Base 5 — Upper bound on s

There is no (57, 106, 2784)-net in base 5, because

1 times m-reduction [i] would yield (57, 105, 2784)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 24 772124 101295 248808 499701 600392 346192 316360 691232 995108 750791 860307 431425 > 5¹⁰⁵ [i]