MinT - Best Known (109−51, 109, s)-Nets in Base 5

Best Known (109−51, 109, s)-Nets in Base 5

(109−51, 109, 104)-Net over F₅ — Constructive and digital

Digital (58, 109, 104)-net over F₅, using

1 times m-reduction [i] based on digital (58, 110, 104)-net over F₅, using
- trace code for nets [i] based on digital (3, 55, 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]

(109−51, 109, 140)-Net over F₅ — Digital

Digital (58, 109, 140)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−51, 109, 2643)-Net in Base 5 — Upper bound on s

There is no (58, 109, 2644)-net in base 5, because

1 times m-reduction [i] would yield (58, 108, 2644)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3103 788827 901760 266180 031342 779362 458880 548186 013693 056711 140897 886689 649745 > 5¹⁰⁸ [i]