MinT - Best Known (72−40, 72, s)-Nets in Base 27

Best Known (72−40, 72, s)-Nets in Base 27

(72−40, 72, 152)-Net over F₂₇ — Constructive and digital

Digital (32, 72, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 26, 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, 46, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(72−40, 72, 224)-Net in Base 27 — Constructive

(32, 72, 224)-net in base 27, using

4 times m-reduction [i] based on (32, 76, 224)-net in base 27, using
- base change [i] based on digital (13, 57, 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]

(72−40, 72, 258)-Net over F₂₇ — Digital

Digital (32, 72, 258)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−40, 72, 298)-Net in Base 27

(32, 72, 298)-net in base 27, using

8 times m-reduction [i] based on (32, 80, 298)-net in base 27, using
- base change [i] based on digital (12, 60, 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]

(72−40, 72, 45409)-Net in Base 27 — Upper bound on s

There is no (32, 72, 45410)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 11 437637 404391 902082 517351 884246 281541 912352 096041 921800 194630 303269 235062 666637 616359 471724 112743 406793 > 27⁷² [i]