MinT - Best Known (37−25, 37, s)-Nets in Base 27

Best Known (37−25, 37, s)-Nets in Base 27

(37−25, 37, 96)-Net over F₂₇ — Constructive and digital

Digital (12, 37, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 37, 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]

(37−25, 37, 109)-Net over F₂₇ — Digital

Digital (12, 37, 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]

(37−25, 37, 116)-Net in Base 27 — Constructive

(12, 37, 116)-net in base 27, using

3 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]

(37−25, 37, 118)-Net in Base 27

(12, 37, 118)-net in base 27, using

3 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]

(37−25, 37, 3997)-Net in Base 27 — Upper bound on s

There is no (12, 37, 3998)-net in base 27, because

1 times m-reduction [i] would yield (12, 36, 3998)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3382 354964 376815 361470 322618 840365 479928 325872 811993 > 27³⁶ [i]