MinT - Best Known (70−27, 70, s)-Nets in Base 27

Best Known (70−27, 70, s)-Nets in Base 27

(70−27, 70, 248)-Net over F₂₇ — Constructive and digital

Digital (43, 70, 248)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 9, 56)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 6, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (4, 13, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
3. digital (4, 17, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (4, 31, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(70−27, 70, 370)-Net in Base 27 — Constructive

(43, 70, 370)-net in base 27, using

38 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(70−27, 70, 2910)-Net over F₂₇ — Digital

Digital (43, 70, 2910)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−27, 70, large)-Net in Base 27 — Upper bound on s

There is no (43, 70, large)-net in base 27, because

25 times m-reduction [i] would yield (43, 45, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]