MinT - Best Known (30, 68, s)-Nets in Base 27

Best Known (30, 68, s)-Nets in Base 27

(30, 68, 146)-Net over F₂₇ — Constructive and digital

Digital (30, 68, 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, 45, 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, 68, 224)-Net in Base 27 — Constructive

(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, 68, 239)-Net over F₂₇ — Digital

Digital (30, 68, 239)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 68, 298)-Net in Base 27

(30, 68, 298)-net in base 27, using

4 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, 68, 40449)-Net in Base 27 — Upper bound on s

There is no (30, 68, 40450)-net in base 27, because

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]