MinT - Best Known (61−33, 61, s)-Nets in Base 27

Best Known (61−33, 61, s)-Nets in Base 27

(61−33, 61, 152)-Net over F₂₇ — Constructive and digital

Digital (28, 61, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 22, 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]
2. digital (6, 39, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(61−33, 61, 172)-Net in Base 27 — Constructive

(28, 61, 172)-net in base 27, using

23 times m-reduction [i] based on (28, 84, 172)-net in base 27, using
- base change [i] based on digital (7, 63, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(61−33, 61, 269)-Net over F₂₇ — Digital

Digital (28, 61, 269)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(61−33, 61, 298)-Net in Base 27

(28, 61, 298)-net in base 27, using

3 times m-reduction [i] based on (28, 64, 298)-net in base 27, using
- base change [i] based on digital (12, 48, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(61−33, 61, 60970)-Net in Base 27 — Upper bound on s

There is no (28, 61, 60971)-net in base 27, because

1 times m-reduction [i] would yield (28, 60, 60971)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 76 181910 923117 860179 665725 762018 866439 649178 088787 854350 028768 827953 956900 806740 364897 > 27⁶⁰ [i]