MinT - Best Known (18, 18+39, s)-Nets in Base 27

Best Known (18, 18+39, s)-Nets in Base 27

(18, 18+39, 108)-Net over F₂₇ — Constructive and digital

Digital (18, 57, 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+39, 116)-Net in Base 27 — Constructive

(18, 57, 116)-net in base 27, using

7 times m-reduction [i] based on (18, 64, 116)-net in base 27, using
- base change [i] based on digital (2, 48, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(18, 18+39, 148)-Net over F₂₇ — Digital

Digital (18, 57, 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+39, 5036)-Net in Base 27 — Upper bound on s

There is no (18, 57, 5037)-net in base 27, because

1 times m-reduction [i] would yield (18, 56, 5037)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 143 411678 774279 833265 337920 451362 241983 725834 531265 272071 968577 472032 530374 244019 > 27⁵⁶ [i]