MinT - Best Known (18, 18+33, s)-Nets in Base 27

Best Known (18, 18+33, s)-Nets in Base 27

(18, 18+33, 108)-Net over F₂₇ — Constructive and digital

Digital (18, 51, 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]

(18, 18+33, 148)-Net over F₂₇ — Digital

Digital (18, 51, 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]

(18, 18+33, 160)-Net in Base 27 — Constructive

(18, 51, 160)-net in base 27, using

1 times m-reduction [i] based on (18, 52, 160)-net in base 27, using
- base change [i] based on digital (5, 39, 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]

(18, 18+33, 167)-Net in Base 27

(18, 51, 167)-net in base 27, using

1 times m-reduction [i] based on (18, 52, 167)-net in base 27, using
- base change [i] based on digital (5, 39, 167)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
      - a curve from the manYPoints database [i]

(18, 18+33, 7764)-Net in Base 27 — Upper bound on s

There is no (18, 51, 7765)-net in base 27, because

1 times m-reduction [i] would yield (18, 50, 7765)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 370232 067386 630173 530452 941852 616292 608098 745821 886908 453499 450269 293633 > 27⁵⁰ [i]