MinT - Best Known (109−24, 109, s)-Nets in Base 5

Best Known (109−24, 109, s)-Nets in Base 5

(109−24, 109, 416)-Net over F₅ — Constructive and digital

Digital (85, 109, 416)-net over F₅, using

1 times m-reduction [i] based on digital (85, 110, 416)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (30, 42, 208)-net over F₅, using
    - trace code for nets [i] based on digital (9, 21, 104)-net over F₂₅, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
        a function field by GarcÃa and Quoos [i]
  2. digital (43, 68, 208)-net over F₅, using
    - trace code for nets [i] based on digital (9, 34, 104)-net over F₂₅, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₂₅ (see above)

(109−24, 109, 4852)-Net over F₅ — Digital

Digital (85, 109, 4852)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−24, 109, 2953096)-Net in Base 5 — Upper bound on s

There is no (85, 109, 2953097)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 15407 439863 822398 511756 544632 752952 230510 668270 995432 795068 632552 641581 752145 > 5¹⁰⁹ [i]