MinT - Best Known (50−23, 50, s)-Nets in Base 5

Best Known (50−23, 50, s)-Nets in Base 5

(50−23, 50, 56)-Net over F₅ — Constructive and digital

Digital (27, 50, 56)-net over F₅, using

trace code for nets [i] based on digital (2, 25, 28)-net over F₂₅, using
- net from sequence [i] based on digital (2, 27)-sequence over F₂₅, using
  - Niederreiter sequence [i]

(50−23, 50, 92)-Net over F₅ — Digital

Digital (27, 50, 92)-net over F₅, using

trace code for nets [i] based on digital (2, 25, 46)-net over F₂₅, using
- net from sequence [i] based on digital (2, 45)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 2 and N(F) ≥ 46, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(50−23, 50, 1586)-Net in Base 5 — Upper bound on s

There is no (27, 50, 1587)-net in base 5, because

1 times m-reduction [i] would yield (27, 49, 1587)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 17827 125360 767542 914879 345221 781349 > 5⁴⁹ [i]