MinT - Best Known (31−12, 31, s)-Nets in Base 25

Best Known (31−12, 31, s)-Nets in Base 25

(31−12, 31, 208)-Net over F₂₅ — Constructive and digital

Digital (19, 31, 208)-net over F₂₅, using

1 times m-reduction [i] based on digital (19, 32, 208)-net over F₂₅, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 1, 26)-net over F₂₅, using
    - s-reduction based on digital (0, 1, s)-net over F₂₅ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  2. digital (0, 1, 26)-net over F₂₅ (see above)
  3. digital (0, 2, 26)-net over F₂₅, using
    - kÂ =Â 2 construction [i]
  4. digital (0, 2, 26)-net over F₂₅ (see above)
  5. digital (0, 3, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
      - Niederreiter sequence [i]
  6. digital (0, 4, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
  7. digital (0, 6, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
  8. digital (0, 13, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)

(31−12, 31, 1786)-Net over F₂₅ — Digital

Digital (19, 31, 1786)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(31−12, 31, 2083052)-Net in Base 25 — Upper bound on s

There is no (19, 31, 2083053)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 21 684045 708403 783856 959694 365102 790024 755345 > 25³¹ [i]