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

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

(39−27, 39, 96)-Net over F₂₇ — Constructive and digital

Digital (12, 39, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 39, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(39−27, 39, 109)-Net over F₂₇ — Digital

Digital (12, 39, 109)-net over F₂₇, using

net from sequence [i] based on digital (12, 108)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 12 and N(F) ≥ 109, using
  - a curve from the manYPoints database [i]

(39−27, 39, 116)-Net in Base 27 — Constructive

(12, 39, 116)-net in base 27, using

1 times m-reduction [i] based on (12, 40, 116)-net in base 27, using
- base change [i] based on digital (2, 30, 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]

(39−27, 39, 118)-Net in Base 27

(12, 39, 118)-net in base 27, using

1 times m-reduction [i] based on (12, 40, 118)-net in base 27, using
- base change [i] based on digital (2, 30, 118)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 117)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 118, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(39−27, 39, 3323)-Net in Base 27 — Upper bound on s

There is no (12, 39, 3324)-net in base 27, because

1 times m-reduction [i] would yield (12, 38, 3324)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2 471671 525492 674388 931072 627650 712359 735859 456749 736313 > 27³⁸ [i]