MinT - Best Known (29−17, 29, s)-Nets in Base 16

Best Known (29−17, 29, s)-Nets in Base 16

(29−17, 29, 66)-Net over F₁₆ — Constructive and digital

Digital (12, 29, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (2, 19, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(29−17, 29, 80)-Net in Base 16 — Constructive

(12, 29, 80)-net in base 16, using

4 times m-reduction [i] based on (12, 33, 80)-net in base 16, using
- base change [i] based on digital (1, 22, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(29−17, 29, 88)-Net over F₁₆ — Digital

Digital (12, 29, 88)-net over F₁₆, using

net from sequence [i] based on digital (12, 87)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 12 and N(F) ≥ 88, using
  - a curve from the manYPoints database [i]

(29−17, 29, 97)-Net in Base 16

(12, 29, 97)-net in base 16, using

1 times m-reduction [i] based on (12, 30, 97)-net in base 16, using
- base change [i] based on digital (2, 20, 97)-net over F₆₄, using
  - net from sequence [i] based on digital (2, 96)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 2 and N(F) ≥ 97, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(29−17, 29, 4107)-Net in Base 16 — Upper bound on s

There is no (12, 29, 4108)-net in base 16, because

1 times m-reduction [i] would yield (12, 28, 4108)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5195 909915 842787 421771 450841 492261 > 16²⁸ [i]