MinT - Best Known (53, 86, s)-Nets in Base 27

Best Known (53, 86, s)-Nets in Base 27

(53, 86, 268)-Net over F₂₇ — Constructive and digital

Digital (53, 86, 268)-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 (6, 39, 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]

(53, 86, 434)-Net in Base 27 — Constructive

(53, 86, 434)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (4, 20, 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. (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]

(53, 86, 3473)-Net over F₂₇ — Digital

Digital (53, 86, 3473)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 86, large)-Net in Base 27 — Upper bound on s

There is no (53, 86, large)-net in base 27, because

31 times m-reduction [i] would yield (53, 55, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]