MinT - Best Known (89−34, 89, s)-Nets in Base 27

Best Known (89−34, 89, s)-Nets in Base 27

(89−34, 89, 274)-Net over F₂₇ — Constructive and digital

Digital (55, 89, 274)-net over F₂₇, using

1 times m-reduction [i] based on digital (55, 90, 274)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 12, 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]
  2. digital (4, 15, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (4, 21, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  4. digital (7, 42, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
        a function field by SÃ©mirat [i]

(89−34, 89, 438)-Net in Base 27 — Constructive

(55, 89, 438)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (5, 22, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]
2. (33, 67, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (33, 68, 370)-net in base 27, using
    - base change [i] based on digital (16, 51, 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]

(89−34, 89, 3687)-Net over F₂₇ — Digital

Digital (55, 89, 3687)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−34, 89, large)-Net in Base 27 — Upper bound on s

There is no (55, 89, large)-net in base 27, because

32 times m-reduction [i] would yield (55, 57, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]