MinT - Best Known (30−12, 30, s)-Nets in Base 25

Best Known (30−12, 30, s)-Nets in Base 25

(30−12, 30, 182)-Net over F₂₅ — Constructive and digital

Digital (18, 30, 182)-net over F₂₅, using

1 times m-reduction [i] based on digital (18, 31, 182)-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, 2, 26)-net over F₂₅, using
    - kÂ =Â 2 construction [i]
  3. digital (0, 2, 26)-net over F₂₅ (see above)
  4. 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]
  5. digital (0, 4, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
  6. digital (0, 6, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
  7. digital (0, 13, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)

(30−12, 30, 1334)-Net over F₂₅ — Digital

Digital (18, 30, 1334)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(30−12, 30, 1218175)-Net in Base 25 — Upper bound on s

There is no (18, 30, 1218176)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 867363 355446 347451 193379 701895 781861 709825 > 25³⁰ [i]