MinT - Best Known (65−40, 65, s)-Nets in Base 16

Best Known (65−40, 65, s)-Nets in Base 16

(65−40, 65, 71)-Net over F₁₆ — Constructive and digital

Digital (25, 65, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 22, 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 (3, 43, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(65−40, 65, 120)-Net in Base 16 — Constructive

(25, 65, 120)-net in base 16, using

5 times m-reduction [i] based on (25, 70, 120)-net in base 16, using
- base change [i] based on digital (11, 56, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(65−40, 65, 144)-Net over F₁₆ — Digital

Digital (25, 65, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(65−40, 65, 4524)-Net in Base 16 — Upper bound on s

There is no (25, 65, 4525)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 857075 733965 190378 784892 401903 882058 968948 612219 819488 229293 579517 613614 548126 > 16⁶⁵ [i]