MinT - Best Known (27, 27+21, s)-Nets in Base 5

Best Known (27, 27+21, s)-Nets in Base 5

(27, 27+21, 104)-Net over F₅ — Constructive and digital

Digital (27, 48, 104)-net over F₅, using

trace code for nets [i] based on digital (3, 24, 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]

(27, 27+21, 112)-Net over F₅ — Digital

Digital (27, 48, 112)-net over F₅, using

trace code for nets [i] based on digital (3, 24, 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]

(27, 27+21, 2176)-Net in Base 5 — Upper bound on s

There is no (27, 48, 2177)-net in base 5, because

1 times m-reduction [i] would yield (27, 47, 2177)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 713 422306 643265 589471 669432 644345 > 5⁴⁷ [i]