MinT - Best Known (54−34, 54, s)-Nets in Base 27

Best Known (54−34, 54, s)-Nets in Base 27

(54−34, 54, 108)-Net over F₂₇ — Constructive and digital

Digital (20, 54, 108)-net over F₂₇, using

t-expansion [i] based on 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]

(54−34, 54, 148)-Net over F₂₇ — Digital

Digital (20, 54, 148)-net over F₂₇, using

t-expansion [i] based on 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]

(54−34, 54, 160)-Net in Base 27 — Constructive

(20, 54, 160)-net in base 27, using

6 times m-reduction [i] based on (20, 60, 160)-net in base 27, using
- base change [i] based on digital (5, 45, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(54−34, 54, 190)-Net in Base 27

(20, 54, 190)-net in base 27, using

2 times m-reduction [i] based on (20, 56, 190)-net in base 27, using
- base change [i] based on digital (6, 42, 190)-net over F₈₁, using
  - net from sequence [i] based on digital (6, 189)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
      - a curve from the manYPoints database [i]

(54−34, 54, 9711)-Net in Base 27 — Upper bound on s

There is no (20, 54, 9712)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 196924 688224 091422 552148 174983 760109 861078 312202 522581 767103 009225 451133 729889 > 27⁵⁴ [i]