MinT - Best Known (19, 19+25, s)-Nets in Base 16

Best Known (19, 19+25, s)-Nets in Base 16

(19, 19+25, 89)-Net over F₁₆ — Constructive and digital

Digital (19, 44, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 31, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(19, 19+25, 104)-Net in Base 16 — Constructive

(19, 44, 104)-net in base 16, using

6 times m-reduction [i] based on (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+25, 129)-Net over F₁₆ — Digital

Digital (19, 44, 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+25, 7272)-Net in Base 16 — Upper bound on s

There is no (19, 44, 7273)-net in base 16, because

1 times m-reduction [i] would yield (19, 43, 7273)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5995 598135 175046 201727 817067 132600 462244 913508 554616 > 16⁴³ [i]