MinT - Best Known (90−52, 90, s)-Nets in Base 27

Best Known (90−52, 90, s)-Nets in Base 27

(90−52, 90, 152)-Net over F₂₇ — Constructive and digital

Digital (38, 90, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 32, 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, 58, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(90−52, 90, 224)-Net in Base 27 — Constructive

(38, 90, 224)-net in base 27, using

10 times m-reduction [i] based on (38, 100, 224)-net in base 27, using
- base change [i] based on digital (13, 75, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(90−52, 90, 246)-Net over F₂₇ — Digital

Digital (38, 90, 246)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−52, 90, 298)-Net in Base 27

(38, 90, 298)-net in base 27, using

14 times m-reduction [i] based on (38, 104, 298)-net in base 27, using
- base change [i] based on digital (12, 78, 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]

(90−52, 90, 36549)-Net in Base 27 — Upper bound on s

There is no (38, 90, 36550)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 665 165123 369666 978786 372186 424893 365932 656947 521857 788433 804208 239504 324059 484006 127702 946825 085027 418580 941452 357233 223146 869213 > 27⁹⁰ [i]