MinT - Best Known (69−38, 69, s)-Nets in Base 27

Best Known (69−38, 69, s)-Nets in Base 27

(69−38, 69, 152)-Net over F₂₇ — Constructive and digital

Digital (31, 69, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 25, 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, 44, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(69−38, 69, 224)-Net in Base 27 — Constructive

(31, 69, 224)-net in base 27, using

3 times m-reduction [i] based on (31, 72, 224)-net in base 27, using
- base change [i] based on digital (13, 54, 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]

(69−38, 69, 263)-Net over F₂₇ — Digital

Digital (31, 69, 263)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−38, 69, 298)-Net in Base 27

(31, 69, 298)-net in base 27, using

7 times m-reduction [i] based on (31, 76, 298)-net in base 27, using
- base change [i] based on digital (12, 57, 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]

(69−38, 69, 48113)-Net in Base 27 — Upper bound on s

There is no (31, 69, 48114)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 581 065904 558357 606239 486054 429454 857580 956333 870010 097840 956919 861184 826966 123578 306629 717228 087841 > 27⁶⁹ [i]