MinT - Best Known (32−15, 32, s)-Nets in Base 5

Best Known (32−15, 32, s)-Nets in Base 5

(32−15, 32, 54)-Net over F₅ — Constructive and digital

Digital (17, 32, 54)-net over F₅, using

trace code for nets [i] based on digital (1, 16, 27)-net over F₂₅, using
- net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
  - Niederreiter sequence [i]

(32−15, 32, 72)-Net over F₅ — Digital

Digital (17, 32, 72)-net over F₅, using

trace code for nets [i] based on digital (1, 16, 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]

(32−15, 32, 1047)-Net in Base 5 — Upper bound on s

There is no (17, 32, 1048)-net in base 5, because

1 times m-reduction [i] would yield (17, 31, 1048)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4658 463623 553778 074721 > 5³¹ [i]