MinT - Best Known (19, 19+31, s)-Nets in Base 16

Best Known (19, 19+31, s)-Nets in Base 16

(19, 19+31, 66)-Net over F₁₆ — Constructive and digital

Digital (19, 50, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 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, 33, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(19, 19+31, 104)-Net in Base 16 — Constructive

(19, 50, 104)-net in base 16, using

base change [i] based on digital (9, 40, 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]

(19, 19+31, 129)-Net over F₁₆ — Digital

Digital (19, 50, 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]

(19, 19+31, 3665)-Net in Base 16 — Upper bound on s

There is no (19, 50, 3666)-net in base 16, because

1 times m-reduction [i] would yield (19, 49, 3666)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 100449 655395 025264 004072 087765 545929 884396 824440 613262 685976 > 16⁴⁹ [i]