MinT - Best Known (37, 37+28, s)-Nets in Base 27

Best Known (37, 37+28, s)-Nets in Base 27

(37, 37+28, 204)-Net over F₂₇ — Constructive and digital

Digital (37, 65, 204)-net over F₂₇, using

1 times m-reduction [i] based on digital (37, 66, 204)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 13, 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, 18, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (6, 35, 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]

(37, 37+28, 370)-Net in Base 27 — Constructive

(37, 65, 370)-net in base 27, using

19 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
- base change [i] based on digital (16, 63, 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]

(37, 37+28, 1187)-Net over F₂₇ — Digital

Digital (37, 65, 1187)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(37, 37+28, 1028250)-Net in Base 27 — Upper bound on s

There is no (37, 65, 1028251)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1093 070329 413538 691829 748133 667159 268213 813576 680199 799920 049186 772199 691785 697322 114564 656109 > 27⁶⁵ [i]