MinT - Best Known (99−25, 99, s)-Nets in Base 5

Best Known (99−25, 99, s)-Nets in Base 5

(99−25, 99, 312)-Net over F₅ — Constructive and digital

Digital (74, 99, 312)-net over F₅, using

5¹ times duplication [i] based on digital (73, 98, 312)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 30, 104)-net over F₅, using
    - trace code for nets [i] based on digital (3, 15, 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]
  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, 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]

(99−25, 99, 1885)-Net over F₅ — Digital

Digital (74, 99, 1885)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−25, 99, 675384)-Net in Base 5 — Upper bound on s

There is no (74, 99, 675385)-net in base 5, because

1 times m-reduction [i] would yield (74, 98, 675385)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 315 548014 240317 678148 836919 662590 040992 189450 007552 348417 307507 238225 > 5⁹⁸ [i]