MinT - Best Known (33−22, 33, s)-Nets in Base 27

Best Known (33−22, 33, s)-Nets in Base 27

(33−22, 33, 96)-Net over F₂₇ — Constructive and digital

Digital (11, 33, 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]

(33−22, 33, 100)-Net over F₂₇ — Digital

Digital (11, 33, 100)-net over F₂₇, using

net from sequence [i] based on digital (11, 99)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 100, using
  - a curve from the manYPoints database [i]

(33−22, 33, 116)-Net in Base 27 — Constructive

(11, 33, 116)-net in base 27, using

3 times m-reduction [i] based on (11, 36, 116)-net in base 27, using
- base change [i] based on digital (2, 27, 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]

(33−22, 33, 118)-Net in Base 27

(11, 33, 118)-net in base 27, using

3 times m-reduction [i] based on (11, 36, 118)-net in base 27, using
- base change [i] based on digital (2, 27, 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]

(33−22, 33, 3711)-Net in Base 27 — Upper bound on s

There is no (11, 33, 3712)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 172270 343296 274164 354234 465218 112536 217572 873729 > 27³³ [i]