MinT - Best Known (32, 32+23, s)-Nets in Base 27

Best Known (32, 32+23, s)-Nets in Base 27

(32, 32+23, 204)-Net over F₂₇ — Constructive and digital

Digital (32, 55, 204)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 11, 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 (6, 29, 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]

(32, 32+23, 370)-Net in Base 27 — Constructive

(32, 55, 370)-net in base 27, using

9 times m-reduction [i] based on (32, 64, 370)-net in base 27, using
- base change [i] based on digital (16, 48, 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]

(32, 32+23, 1330)-Net over F₂₇ — Digital

Digital (32, 55, 1330)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(32, 32+23, 2007864)-Net in Base 27 — Upper bound on s

There is no (32, 55, 2007865)-net in base 27, because

1 times m-reduction [i] would yield (32, 54, 2007865)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 196627 209585 142918 994656 791527 798479 866892 432742 487797 766189 093080 028403 242723 > 27⁵⁴ [i]