MinT - Best Known (112−25, 112, s)-Nets in Base 5

Best Known (112−25, 112, s)-Nets in Base 5

(112−25, 112, 460)-Net over F₅ — Constructive and digital

Digital (87, 112, 460)-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 (45, 70, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 35, 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]

(112−25, 112, 4491)-Net over F₅ — Digital

Digital (87, 112, 4491)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−25, 112, 3861650)-Net in Base 5 — Upper bound on s

There is no (87, 112, 3861651)-net in base 5, because

1 times m-reduction [i] would yield (87, 111, 3861651)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 385186 148166 792212 809475 209714 002542 317084 294240 861966 469254 983713 781590 777425 > 5¹¹¹ [i]