MinT - Best Known (44−19, 44, s)-Nets in Base 5

Best Known (44−19, 44, s)-Nets in Base 5

(44−19, 44, 104)-Net over F₅ — Constructive and digital

Digital (25, 44, 104)-net over F₅, using

trace code for nets [i] based on digital (3, 22, 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]

(44−19, 44, 112)-Net over F₅ — Digital

Digital (25, 44, 112)-net over F₅, using

trace code for nets [i] based on digital (3, 22, 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]

(44−19, 44, 2259)-Net in Base 5 — Upper bound on s

There is no (25, 44, 2260)-net in base 5, because

1 times m-reduction [i] would yield (25, 43, 2260)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 139032 609065 220314 263308 039249 > 5⁴³ [i]