MinT - Best Known (26−15, 26, s)-Nets in Base 16

Best Known (26−15, 26, s)-Nets in Base 16

(26−15, 26, 66)-Net over F₁₆ — Constructive and digital

Digital (11, 26, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 9, 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, 17, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(26−15, 26, 80)-Net in Base 16 — Constructive

(11, 26, 80)-net in base 16, using

4 times m-reduction [i] based on (11, 30, 80)-net in base 16, using
- base change [i] based on digital (1, 20, 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]

(26−15, 26, 81)-Net over F₁₆ — Digital

Digital (11, 26, 81)-net over F₁₆, using

t-expansion [i] based on digital (10, 26, 81)-net over F₁₆, using
- net from sequence [i] based on digital (10, 80)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 10 and N(F) ≥ 81, using
    - a curve from the manYPoints database [i]

(26−15, 26, 97)-Net in Base 16

(11, 26, 97)-net in base 16, using

1 times m-reduction [i] based on (11, 27, 97)-net in base 16, using
- base change [i] based on digital (2, 18, 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]

(26−15, 26, 4496)-Net in Base 16 — Upper bound on s

There is no (11, 26, 4497)-net in base 16, because

1 times m-reduction [i] would yield (11, 25, 4497)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 267682 801683 998385 821011 430736 > 16²⁵ [i]