MinT - Best Known (41−15, 41, s)-Nets in Base 25

Best Known (41−15, 41, s)-Nets in Base 25

(41−15, 41, 286)-Net over F₂₅ — Constructive and digital

Digital (26, 41, 286)-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, 1, 26)-net over F₂₅ (see above)
4. digital (0, 1, 26)-net over F₂₅ (see above)
5. digital (0, 2, 26)-net over F₂₅, using
  - kÂ =Â 2 construction [i]
6. digital (0, 2, 26)-net over F₂₅ (see above)
7. 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]
8. digital (0, 3, 26)-net over F₂₅ (see above)
9. digital (0, 5, 26)-net over F₂₅, using
  - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
10. digital (0, 7, 26)-net over F₂₅, using
  - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
11. digital (0, 15, 26)-net over F₂₅, using
  - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)

(41−15, 41, 3135)-Net over F₂₅ — Digital

Digital (26, 41, 3135)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(41−15, 41, large)-Net in Base 25 — Upper bound on s

There is no (26, 41, large)-net in base 25, because

13 times m-reduction [i] would yield (26, 28, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]