MinT - Best Known (10, 10+13, s)-Nets in Base 16

Best Known (10, 10+13, s)-Nets in Base 16

(10, 10+13, 66)-Net over F₁₆ — Constructive and digital

Digital (10, 23, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 8, 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, 15, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(10, 10+13, 80)-Net in Base 16 — Constructive

(10, 23, 80)-net in base 16, using

4 times m-reduction [i] based on (10, 27, 80)-net in base 16, using
- base change [i] based on digital (1, 18, 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]

(10, 10+13, 81)-Net over F₁₆ — Digital

Digital (10, 23, 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]

(10, 10+13, 97)-Net in Base 16

(10, 23, 97)-net in base 16, using

1 times m-reduction [i] based on (10, 24, 97)-net in base 16, using
- base change [i] based on digital (2, 16, 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]

(10, 10+13, 5187)-Net in Base 16 — Upper bound on s

There is no (10, 23, 5188)-net in base 16, because

1 times m-reduction [i] would yield (10, 22, 5188)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 309 507223 080000 296934 605296 > 16²² [i]