MinT - Best Known (54, 54+37, s)-Nets in Base 25

Best Known (54, 54+37, s)-Nets in Base 25

(54, 54+37, 326)-Net over F₂₅ — Constructive and digital

Digital (54, 91, 326)-net over F₂₅, using

1 times m-reduction [i] based on digital (54, 92, 326)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 29, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]
  2. digital (25, 63, 200)-net over F₂₅, using
    - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
        a function field by GarcÃa and Quoos [i]

(54, 54+37, 2051)-Net over F₂₅ — Digital

Digital (54, 91, 2051)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+37, 3073389)-Net in Base 25 — Upper bound on s

There is no (54, 91, 3073390)-net in base 25, because

1 times m-reduction [i] would yield (54, 90, 3073390)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 652533 763218 752009 396007 667501 226360 531184 253498 744979 065806 363110 623879 263075 971892 410237 878188 657702 305234 591178 990995 757025 > 25⁹⁰ [i]