MinT - Best Known (87−33, 87, s)-Nets in Base 27

Best Known (87−33, 87, s)-Nets in Base 27

(87−33, 87, 274)-Net over F₂₇ — Constructive and digital

Digital (54, 87, 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, 20, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (7, 40, 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]

(87−33, 87, 438)-Net in Base 27 — Constructive

(54, 87, 438)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 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, 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]

(87−33, 87, 3848)-Net over F₂₇ — Digital

Digital (54, 87, 3848)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−33, 87, large)-Net in Base 27 — Upper bound on s

There is no (54, 87, large)-net in base 27, because

31 times m-reduction [i] would yield (54, 56, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]