MinT - Best Known (75−27, 75, s)-Nets in Base 81

Best Known (75−27, 75, s)-Nets in Base 81

(75−27, 75, 820)-Net over F₈₁ — Constructive and digital

Digital (48, 75, 820)-net over F₈₁, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 82)-net over F₈₁, using
  - kÂ =Â 2 construction [i]
2. digital (0, 3, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]
3. digital (0, 3, 82)-net over F₈₁ (see above)
4. digital (0, 3, 82)-net over F₈₁ (see above)
5. digital (0, 4, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
6. digital (0, 5, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
7. digital (0, 6, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
8. digital (0, 9, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
9. digital (0, 13, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
10. digital (0, 27, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)

(75−27, 75, 42226)-Net over F₈₁ — Digital

Digital (48, 75, 42226)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−27, 75, large)-Net in Base 81 — Upper bound on s

There is no (48, 75, large)-net in base 81, because

25 times m-reduction [i] would yield (48, 50, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]