MinT - Best Known (49−26, 49, s)-Nets in Base 16

Best Known (49−26, 49, s)-Nets in Base 16

(49−26, 49, 110)-Net over F₁₆ — Constructive and digital

Digital (23, 49, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 32, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(49−26, 49, 150)-Net in Base 16 — Constructive

(23, 49, 150)-net in base 16, using

2 times m-reduction [i] based on (23, 51, 150)-net in base 16, using
- base change [i] based on (6, 34, 150)-net in base 64, using
  - 1 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
    - base change [i] based on digital (1, 30, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(49−26, 49, 158)-Net over F₁₆ — Digital

Digital (23, 49, 158)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(49−26, 49, 172)-Net in Base 16

(23, 49, 172)-net in base 16, using

base change [i] based on digital (2, 28, 172)-net over F₁₂₈, using
- net from sequence [i] based on digital (2, 171)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 2 and N(F) ≥ 172, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(49−26, 49, 13052)-Net in Base 16 — Upper bound on s

There is no (23, 49, 13053)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 100486 864726 756637 895395 830920 694289 185787 519774 953415 943136 > 16⁴⁹ [i]