MinT - Best Known (89−32, 89, s)-Nets in Base 27

Best Known (89−32, 89, s)-Nets in Base 27

(89−32, 89, 298)-Net over F₂₇ — Constructive and digital

Digital (57, 89, 298)-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 (6, 16, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
3. digital (6, 22, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
4. digital (7, 39, 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]

(89−32, 89, 486)-Net in Base 27 — Constructive

(57, 89, 486)-net in base 27, using

1 times m-reduction [i] based on (57, 90, 486)-net in base 27, using
- (u,Â u+v)-construction [i] based on
  1. (8, 24, 116)-net in base 27, using
    - base change [i] based on digital (2, 18, 116)-net over F₈₁, using
      - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]
  2. (33, 66, 370)-net in base 27, using
    - 2 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]

(89−32, 89, 6160)-Net over F₂₇ — Digital

Digital (57, 89, 6160)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−32, 89, large)-Net in Base 27 — Upper bound on s

There is no (57, 89, large)-net in base 27, because

30 times m-reduction [i] would yield (57, 59, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]