MinT - Best Known (35−21, 35, s)-Nets in Base 16

Best Known (35−21, 35, s)-Nets in Base 16

(35−21, 35, 66)-Net over F₁₆ — Constructive and digital

Digital (14, 35, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 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, 23, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(35−21, 35, 97)-Net over F₁₆ — Digital

Digital (14, 35, 97)-net over F₁₆, using

t-expansion [i] based on digital (13, 35, 97)-net over F₁₆, using
- net from sequence [i] based on digital (13, 96)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 13 and N(F) ≥ 97, using
    - a curve from the manYPoints database [i]

(35−21, 35, 98)-Net in Base 16 — Constructive

(14, 35, 98)-net in base 16, using

base change [i] based on digital (7, 28, 98)-net over F₃₂, using
- net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(35−21, 35, 3743)-Net in Base 16 — Upper bound on s

There is no (14, 35, 3744)-net in base 16, because

1 times m-reduction [i] would yield (14, 34, 3744)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 87195 639355 660237 087604 190370 272326 954101 > 16³⁴ [i]