MinT - Best Known (27−12, 27, s)-Nets in Base 5

Best Known (27−12, 27, s)-Nets in Base 5

(27−12, 27, 54)-Net over F₅ — Constructive and digital

Digital (15, 27, 54)-net over F₅, using

1 times m-reduction [i] based on digital (15, 28, 54)-net over F₅, using
- trace code for nets [i] based on digital (1, 14, 27)-net over F₂₅, using
  - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
    - Niederreiter sequence [i]

(27−12, 27, 72)-Net over F₅ — Digital

Digital (15, 27, 72)-net over F₅, using

1 times m-reduction [i] based on digital (15, 28, 72)-net over F₅, using
- trace code for nets [i] based on digital (1, 14, 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]

(27−12, 27, 1041)-Net in Base 5 — Upper bound on s

There is no (15, 27, 1042)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7 450800 634252 645633 > 5²⁷ [i]