MinT - Best Known (54−36, 54, s)-Nets in Base 27

Best Known (54−36, 54, s)-Nets in Base 27

(54−36, 54, 108)-Net over F₂₇ — Constructive and digital

Digital (18, 54, 108)-net over F₂₇, using

net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
  - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(54−36, 54, 148)-Net over F₂₇ — Digital

Digital (18, 54, 148)-net over F₂₇, using

net from sequence [i] based on digital (18, 147)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 148, using
  - a curve from the manYPoints database [i]

(54−36, 54, 150)-Net in Base 27 — Constructive

(18, 54, 150)-net in base 27, using

2 times m-reduction [i] based on (18, 56, 150)-net in base 27, using
- base change [i] based on digital (4, 42, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(54−36, 54, 154)-Net in Base 27

(18, 54, 154)-net in base 27, using

2 times m-reduction [i] based on (18, 56, 154)-net in base 27, using
- base change [i] based on digital (4, 42, 154)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 153)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 154, using
      - a curve from the manYPoints database [i]

(54−36, 54, 5708)-Net in Base 27 — Upper bound on s

There is no (18, 54, 5709)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 196712 605800 702173 968221 099217 631466 688425 314010 417319 310044 145128 114595 300145 > 27⁵⁴ [i]