MinT - Best Known (41−19, 41, s)-Nets in Base 5

Best Known (41−19, 41, s)-Nets in Base 5

(41−19, 41, 54)-Net over F₅ — Constructive and digital

Digital (22, 41, 54)-net over F₅, using

1 times m-reduction [i] based on digital (22, 42, 54)-net over F₅, using
- trace code for nets [i] based on digital (1, 21, 27)-net over F₂₅, using
  - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
    - Niederreiter sequence [i]

(41−19, 41, 72)-Net over F₅ — Digital

Digital (22, 41, 72)-net over F₅, using

1 times m-reduction [i] based on digital (22, 42, 72)-net over F₅, using
- trace code for nets [i] based on digital (1, 21, 36)-net over F₂₅, using
  - net from sequence [i] based on digital (1, 35)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 1 and N(F) ≥ 36, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(41−19, 41, 1318)-Net in Base 5 — Upper bound on s

There is no (22, 41, 1319)-net in base 5, because

1 times m-reduction [i] would yield (22, 40, 1319)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 9108 037607 387175 321170 723965 > 5⁴⁰ [i]