MinT - Best Known (30, 30+39, s)-Nets in Base 27

Best Known (30, 30+39, s)-Nets in Base 27

(30, 30+39, 146)-Net over F₂₇ — Constructive and digital

Digital (30, 69, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (7, 46, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(30, 30+39, 172)-Net in Base 27 — Constructive

(30, 69, 172)-net in base 27, using

23 times m-reduction [i] based on (30, 92, 172)-net in base 27, using
- base change [i] based on digital (7, 69, 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]

(30, 30+39, 227)-Net over F₂₇ — Digital

Digital (30, 69, 227)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 30+39, 298)-Net in Base 27

(30, 69, 298)-net in base 27, using

3 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+39, 40449)-Net in Base 27 — Upper bound on s

There is no (30, 69, 40450)-net in base 27, because

1 times m-reduction [i] would yield (30, 68, 40450)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 519672 721077 733707 967075 899780 258685 005742 667539 810026 607947 585327 913956 605141 512728 869196 280161 > 27⁶⁸ [i]