MinT - Best Known (56, 56+38, s)-Nets in Base 27

Best Known (56, 56+38, s)-Nets in Base 27

(56, 56+38, 260)-Net over F₂₇ — Constructive and digital

Digital (56, 94, 260)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 19, 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]
2. digital (8, 27, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]
3. digital (10, 48, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(56, 56+38, 422)-Net in Base 27 — Constructive

(56, 94, 422)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (3, 22, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]
2. (34, 72, 370)-net in base 27, using
  - base change [i] based on digital (16, 54, 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]

(56, 56+38, 2458)-Net over F₂₇ — Digital

Digital (56, 94, 2458)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+38, 3678958)-Net in Base 27 — Upper bound on s

There is no (56, 94, 3678959)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 353 341643 905388 470617 180935 559082 045413 859168 224639 861057 305577 285215 845224 227836 204741 792454 187176 587622 482112 566919 234421 298573 220435 > 27⁹⁴ [i]