MinT - Best Known (49−21, 49, s)-Nets in Base 5

Best Known (49−21, 49, s)-Nets in Base 5

(49−21, 49, 104)-Net over F₅ — Constructive and digital

Digital (28, 49, 104)-net over F₅, using

1 times m-reduction [i] based on digital (28, 50, 104)-net over F₅, using
- trace code for nets [i] based on digital (3, 25, 52)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by GarcÃa and Quoos [i]

(49−21, 49, 112)-Net over F₅ — Digital

Digital (28, 49, 112)-net over F₅, using

1 times m-reduction [i] based on digital (28, 50, 112)-net over F₅, using
- trace code for nets [i] based on digital (3, 25, 56)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 55)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 56, using
      - fields by Serre with genus 3 or 4 [i]

(49−21, 49, 2557)-Net in Base 5 — Upper bound on s

There is no (28, 49, 2558)-net in base 5, because

1 times m-reduction [i] would yield (28, 48, 2558)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3562 038291 038508 044003 489694 245505 > 5⁴⁸ [i]