MinT - Best Known (55−34, 55, s)-Nets in Base 16

Best Known (55−34, 55, s)-Nets in Base 16

(55−34, 55, 66)-Net over F₁₆ — Constructive and digital

Digital (21, 55, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 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, 36, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(55−34, 55, 104)-Net in Base 16 — Constructive

(21, 55, 104)-net in base 16, using

5 times m-reduction [i] based on (21, 60, 104)-net in base 16, using
- base change [i] based on digital (9, 48, 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]

(55−34, 55, 129)-Net over F₁₆ — Digital

Digital (21, 55, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 55, 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]

(55−34, 55, 3753)-Net in Base 16 — Upper bound on s

There is no (21, 55, 3754)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 685834 373929 914720 044927 247591 442233 960144 247887 281037 891269 222646 > 16⁵⁵ [i]