MinT - Best Known (34, 53, s)-Nets in Base 81

Best Known (34, 53, s)-Nets in Base 81

(34, 53, 984)-Net over F₈₁ — Constructive and digital

Digital (34, 53, 984)-net over F₈₁, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 82)-net over F₈₁, using
  - s-reduction based on digital (0, 1, s)-net over F₈₁ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 1, 82)-net over F₈₁ (see above)
3. digital (0, 1, 82)-net over F₈₁ (see above)
4. digital (0, 2, 82)-net over F₈₁, using
  - kÂ =Â 2 construction [i]
5. digital (0, 2, 82)-net over F₈₁ (see above)
6. digital (0, 2, 82)-net over F₈₁ (see above)
7. 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]
8. digital (0, 3, 82)-net over F₈₁ (see above)
9. digital (0, 4, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
10. digital (0, 6, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
11. digital (0, 9, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
12. digital (0, 19, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)

(34, 53, 39316)-Net over F₈₁ — Digital

Digital (34, 53, 39316)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 53, large)-Net in Base 81 — Upper bound on s

There is no (34, 53, large)-net in base 81, because

17 times m-reduction [i] would yield (34, 36, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]