MinT - Best Known (53−33, 53, s)-Nets in Base 16

Best Known (53−33, 53, s)-Nets in Base 16

(53−33, 53, 66)-Net over F₁₆ — Constructive and digital

Digital (20, 53, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 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, 35, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(53−33, 53, 104)-Net in Base 16 — Constructive

(20, 53, 104)-net in base 16, using

2 times m-reduction [i] based on (20, 55, 104)-net in base 16, using
- base change [i] based on digital (9, 44, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(53−33, 53, 129)-Net over F₁₆ — Digital

Digital (20, 53, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 53, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(53−33, 53, 3705)-Net in Base 16 — Upper bound on s

There is no (20, 53, 3706)-net in base 16, because

1 times m-reduction [i] would yield (20, 52, 3706)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 412 440613 430405 129918 868885 573902 847602 847049 700187 760226 558941 > 16⁵² [i]