MinT - Best Known (9, 9+17, s)-Nets in Base 27

Best Known (9, 9+17, s)-Nets in Base 27

(9, 9+17, 88)-Net over F₂₇ — Constructive and digital

Digital (9, 26, 88)-net over F₂₇, using

net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
  - a function field by SÃ©mirat [i]

(9, 9+17, 99)-Net over F₂₇ — Digital

Digital (9, 26, 99)-net over F₂₇, using

net from sequence [i] based on digital (9, 98)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 99, using
  - a curve from the manYPoints database [i]

(9, 9+17, 116)-Net in Base 27 — Constructive

(9, 26, 116)-net in base 27, using

2 times m-reduction [i] based on (9, 28, 116)-net in base 27, using
- base change [i] based on digital (2, 21, 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]

(9, 9+17, 118)-Net in Base 27

(9, 26, 118)-net in base 27, using

2 times m-reduction [i] based on (9, 28, 118)-net in base 27, using
- base change [i] based on digital (2, 21, 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]

(9, 9+17, 4298)-Net in Base 27 — Upper bound on s

There is no (9, 26, 4299)-net in base 27, because

1 times m-reduction [i] would yield (9, 25, 4299)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 608528 676444 166629 854808 639248 407473 > 27²⁵ [i]