MinT - Best Known (30, 30+37, s)-Nets in Base 27

Best Known (30, 30+37, s)-Nets in Base 27

(30, 30+37, 152)-Net over F₂₇ — Constructive and digital

Digital (30, 67, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 24, 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, 43, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(30, 30+37, 224)-Net in Base 27 — Constructive

(30, 67, 224)-net in base 27, using

1 times m-reduction [i] based on (30, 68, 224)-net in base 27, using
- base change [i] based on digital (13, 51, 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]

(30, 30+37, 254)-Net over F₂₇ — Digital

Digital (30, 67, 254)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 30+37, 298)-Net in Base 27

(30, 67, 298)-net in base 27, using

5 times m-reduction [i] based on (30, 72, 298)-net in base 27, using
- base change [i] based on digital (12, 54, 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]

(30, 30+37, 51453)-Net in Base 27 — Upper bound on s

There is no (30, 67, 51454)-net in base 27, because

1 times m-reduction [i] would yield (30, 66, 51454)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 29520 260689 292493 321030 807040 666995 873225 870770 652740 454988 673875 194643 350562 585930 926601 616989 > 27⁶⁶ [i]