MinT - Best Known (36−15, 36, s)-Nets in Base 5

Best Known (36−15, 36, s)-Nets in Base 5

(36−15, 36, 104)-Net over F₅ — Constructive and digital

Digital (21, 36, 104)-net over F₅, using

trace code for nets [i] based on digital (3, 18, 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]

(36−15, 36, 112)-Net over F₅ — Digital

Digital (21, 36, 112)-net over F₅, using

trace code for nets [i] based on digital (3, 18, 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]

(36−15, 36, 2635)-Net in Base 5 — Upper bound on s

There is no (21, 36, 2636)-net in base 5, because

1 times m-reduction [i] would yield (21, 35, 2636)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 911231 166811 994975 715505 > 5³⁵ [i]